LIBRARY 

OF   THE 

UNIVERSITY  OF  CALIFORNIA. 

GIFT  OF" 

MRS.   MARTHA   E.   HALLIDIE. 
Class 


WORKS  OF  PROF.  H.  W.  SPANGLER 


PUBLISHED    BY 


JOHN  WILEY  &  SONS. 


Valve-Gears. 

Designed  as  a  Text-book  giving  those  parts  of  the 
Theory  of  Valve-gears  necessary  to  a  clear  under- 
standing of  the  subject.  8vo,  xii  4-179  pages,  109 
figures,  cloth,  $2.50. 


Notes  on  Thermodynamics. 

The  Derivation  of  the  Fundamental  Principles  of 
Thermodynamics  and  their  Application  to  Numer 
ical  Problems,  ismo,  vi  +  6g  pages,  24  figures, 
cloth,  $1.00. 


NOTES 


ON 


THERMODYNAMICS. 


BY 

H.     W.     SPANGLER, 

Whitney  Professor  of  Mechanical  Engineering 
in  the  University  of  Pennsylvania. 


PART    I. 


SECOND    EDITION-. 
FIRST     THOUSAND. 


JOHN    WILEY    &    SONS. 

LONDON:    CHAPMAN   &    HALL,    LIMITED. 

1901. 


QJ 


— *S 


Copyright,  1901, 

BY 
H.   W.    SPANGLER. 


HALLIDIE 


ROBERT   DRUMMOND,    PRINTER,    NEW   YORK 


PREFACE. 


FOR  the  purpose  of  covering  the  theoretical  side  of 
thermodynamics  more  rapidly  than  could  be  done  with 
the  aid  of  existing  text-books,  the  author  prepared 
these  notes  four  years  ago  for  use  in  his  classes. 

The  results  were  fairly  satisfactory,  and  as  the  work 
is  now  used  by  other  teachers,  a  revised  edition  has 
been  prepared.  In  this,  errors  have  been  corrected, 
the  text  has  been  condensed,  and  additional  problems 
have  been  added. 

It  is  not  intended  as  a  reference-book,  except  for 
those  who  have  worked  it  through  and  have  solved  the 
problems. 

There  is  little  that  is  new  in  it.  All  the  later 
writers  have  been  consulted  in  preparing  the  work, 
and  whatever  has  seemed  the  most  satisfactory  method 
of  arriving  at  a  result  has  been  made  use  of. 

The  work  is  not  complete  in  itself,  and  a  good  table 
of  the  properties  of  vapors  is  required  to  work  out 
many  of  the  problems.  The  tables  prepared  by  Pro- 
fessor Peabody  are  used  in  the  text. 

H.  W.  SPANGLER. 

UNIVERSITY  OF  PENNSYLVANIA, 
June  6,  1901. 

iii 

96050 


NOTATION. 

A  =  Heat  equivalent  of  work  = 

c  =  Specific  heat,  the  subscript  indicating  the  law 
of    the     expansion,    and    is     used    whether 
units  are  foot-pounds  or  heat-units. 
H  —  Heat  required  in  heat-units  or  foot-pounds. 
J  =  Mechanical  equivalent  of  heat  =  778. 
K  =  Constant  of  equation  pvn  =  K. 

A  —  Total  heat  required  to  make  I  pound  of  vapor 

from  liquid  at  32  degrees  F. 
m  =  Weight.' 
M=  Weight. 

n  =  Exponent  in  equation  pvn  =  K. 

p  =  Pressure  in  pounds  per  square  foot,  absolute. 

q  =  Heat  of  liquid. 

r  =  Total  latent  heat. 

p  —  Inner  latent  heat. 
R  —  Constant  for  any  substance  in  equation/^=^  T. 

s  =  Volume  of  I  pound  of  vapor. 

cr  =  Volume  of  I  pound  of  liquid. 

/  —  Temperature  Fahrenheit. 

T=  Temperature  absolute. 

v  —  Difference  between  the  volume  of  I  pound  of 
vapor  and  I  pound  of  liquid  =  s —  cr. 

i)  —  Volume  in  cubic  feet  of  I  pound. 

V  =  Any  volume. 
W  =  Work,  foot-pounds  or  heat-units. 


NOTES  ON  THERMODYNAMICS. 


IN  Physics  a  distinction  is  made  between  perfect 
gases  and  vapors.  In  this  work  we  will  also  deal  with 
these  two  classes  of  substances,  and,  for  engineering 
purposes,  perfect  gases  are  such  as  practically  obey  the 
laws  of  Boyle  and  Charles.  Under  the  head  of  perfect 
gases  would  be  classed  air,  hydrogen,  oxygen,  super- 
heated steam,  ammonia,  carbonic  acid,  etc.,  all  being 
sufficiently  far  from  their  condensing-point  to  obey 
the  laws  referred  to  above. 

In  the  shape  of  a  formula  these  laws  can  be  best 
stated  as 

This  equation  is  constantly  being  used  in  thermody- 
namics, and  the  exact  meaning  of  the  terms  is  impor- 
tant. In  all  this  work  English  units,  pounds,  feet, 
and  degrees  Fahrenheit  will  be  used.  In  these  units 
th»  following  definitions  may  be  given  to  the  terms  of 
equation  (i): 

p  is  the  absolute  pressure  in  pounds  per  square  foot. 

v  is  the  volume  in  cubic  feet  of  I  pound  of  the  sub- 
stance dealt  with. 


NOTES   ON   THERMODYNAMICS. 

T  is  the  absolute  temperature,  Fahrenheit  degrees. 

R  is  a  constant  whose  value  depends  on  the  sub- 
stance and  the  units  taken. 

To  determine  the  value  of  R  for  any  substance,  we 
must  have  for  one  given  condition  of  pressure  and 
temperature  the  corresponding  value  of  the  volume  of 
I  pound.  This  we  have  for  many  substances.  Thus, 
for  air  we  have,  for  a  pressure  of  14.7  pounds  per 
square  inch,  or  /  =  14.7  X  144,  and  a  temperature  of 
32  degrees  Fahrenheit,  or  T  =  492.7,  the  volume  of 
I  pound  of  air,  or  v  =  12.39  cubic  feet.  These  are 
quantities  determined  by  experiment.  Putting  these 
values  in  equation  (i),  we  have  for  air 

K  _    pv         14.7  X   H4  X  12.39   . 
T  492^7 

or,  for  air,  with  the  units  we  have  taken,  we  have 

pv  =  53,37^- 

This  equation  is  always  true  for  air,  and  if,  at  any 
time  or  under  any  conditions,  two  of  the  variables  in 
the  equation  are  given,  the  third  can  be  found. 

Problem  1. — 10  pounds  of  air  at  200  degrees  F.  occupy  120 
cubic  feet;  what  must  be  the  pressure  ? 

Here  T  =  460.7  +  200  =  660.7  ;   v  —  —  =  12; 

10 

53.  vj  x  660.7 
p  =  -  —  =  2950  pounds  per  square  foot.* 

Probi  2. — How  many  pounds  of  air  does  it  take  to  fill  5600  cubic 
feet  at  1 5  pounds  pressure  per  square  inch  and  at  60 
degrees  F.  ? 

*  The  slide  rule  or  three-place  logarithms  are  used  in  the  solu- 
tion of  all  problems,  and  the  result  is  probably  correct  within  2$. 


NOTES   ON    THERMODYNAMICS. 


Here/  =  15  x  144;    T  =  460.7  -f  .60  =  520.7  ; 


_  52°-7  x  53-37  _ 
15x144 

of  i   pound,  5600  cubic  feet   contain 


5600 


voiume 


=   434 


pounds  of  air. 
Prob,  3. — At  what  temperature  will  10  pounds  of  air  at  1 5  pounds 

pressure  per  square  inch  fill  60  cubic  feet? 
Prob.  4. — What  must  be  the  pressure  in  a  vessel  of  4  cubic  feet 

if  it  contains  30  pounds  of  air  at  50  degrees  F.  ?     ^? 

Evidently,  if,  in  equation  (i),  we  are  dealing  with 
a  substance  twice  as  heavy  as  air,  the  value  of  v  in  the 
first  member,  or  the  volume  of  a  pound,  will  be  only 
half  as  great  and,  consequently,  the  value  of  R  would 
be  only  half  as  great. 


Substance.* 

Relative 
Density. 

R 
Value. 

Air 

14.4 

e-7     -17 

CIA 

o 

16 

14.4 

X   '(I  ^7 

48  I 

H 

I 

16 

MA 

X   %1  ^7 

77O 

N 



14 

14^4 

X  53-37 

rOj 

••*X*i 

22 

14 

X  53-37 

NH 

JO 

8  5 

22 
14.4 

X  53-37 

gO   6 

ro 

14 

8-5 

X  53-37 

(steam)           .... 

Q 
9 

14 

X   "^   ^7 

54-9 
85  6 

9 

*  Some  of  these  substances  do  not  act  as  perfect  gases  at  usual 
pressures  and  temperatures,  so  that  care  must  be  exercised  in 
using  these  constants. 


4  NOTES  ON   THERMODYNAMICS. 

This  enables  us  to  apply  the  formula  of  equation  (i) 
and  its  constant,  as  determined  for  air,  to  many  other 
substances.  From  a  table  of  relative  densities  one  can 
readily  determine  the  value  of  R  for  these  substances, 
as  in  the  table  on  page  3,  and  these  values  are  practi- 
cally correct  for  engineering  calculations. 

Prob,  5.  —  How  many  pounds  of  oxygen  will  a  holder  contain 
whose  volume  is  3  cubic  feet,  pressure  250  pounds 
per  square  inch,  and  temperature  75  degrees  F.? 
We  have  for  oxygen 


/  =  250  x  144  =  36000; 

14.4  x  53.37 

^—^  L  x  535-7 

36000  ~  and 

3x36000x16  ^  „ 

14.4x53.37x535.7 

Prob,  6,  —  What  weight  of  hydrogen  will  fill  a  holder  of  3.5 
cubic  feet  at  200  pounds  pressure  and  /  =  80 
degrees  F.?  /  / 

Prob.  7.  —  What  is  the  temperature  at  which  a  cubic  foot  of  CO2 
will  weigh  .2  pound  at  100  pounds  pressure?  /  r 

It  is  convenient  to  reduce  the  expression  for  the 
weight  to  a  simple  formula.  If  V  \s  the  total  volume 
and  v  the  volume  per  pound,  then 

V  pV 

—  •=.  weight,  or  M  =  ^-^y 

V  4\.  1 

from  which  of  course,  if  V  =  v,  the  weight  is  I  pound. 


NOTES  ON   THERMODYNAMICS.  5 

Prob.  8.  —  How   many  pounds  of  air  will   fill   a  vessel  of  400 
cubic  feet  at  15  pounds  pressure  if  one-half  the 
volume  is  at  80  degrees   F.  and  the   rest  at  600 
degrees  F.? 
The  weight  of  the  portion  at  80  degrees  is 

15  x  144  x  200 

— 


53.37  x  541 

and  the  weight  of  the  remainder  is 

15  x  144  x  200 

J/a  =  —  —2  -  =  7-62. 

53.37  x  1061 
The  total  weight  is  22.65  pounds. 

Prob.  9.  —  What  must  be  the  pressure  at  which  20  pounds  of 
air  will  fill  270  cubic  feet,  180  cubic  feet  being  at 
500  degrees  and  90  cubic  feet  at  60  degrees  F.? 

In  defining  a  perfect  gas,  there  was  one  peculiarity 
which  was  not  mentioned  and  which  will  now  be  of 
use.  When  a  perfect  gas  is  allowed  to  remain  at  the 
same  temperature  while  its  volume  changes,  the 
amount  of  heat  that  must  be  added  to  it  to  change 
its  pressure  and  volume  is  that  required  to  do  the 
external  work  and  no  more.  That  is,  if  a  perfect  gas 
is  allowed  to  expand  and  change  its  temperature,  the 
quantity  of  heat  which  must  be  added  to  it  is  that 
required  to  change  the  temperature,  added  to  that 
required  to  do  external  work. 

As  the  equation/^  =  R  T  contains  three  variables, 
it  is  not  convenient  to  indicate  all  the  variations  of 
/>,  v,  and  T  on  the  same  diagram,  and  for  convenience 
of  representation,  and  because  a  diagram  whose  co- 
ordinates are  pressure  and  volume  is  a  diagram  of 
work,  the  /,  v  co-ordinates  will  be  understood  unless 
different  co-ordinates  are  marked  on  the  figure. 


6 


NOTES   ON    THERMODYNAMICS. 


b 

a 

3 

F 

j  VOLUMES,  rc 
IG.    I. 

Thus,  in  Fig.  I  if  we  call  the  two  axes  pressure  and 
volume,  and  we  have  a  pound  of  gas  in  the  conditions 
represented  by  a,  its  volume  is  oy  and 
its  pressure  is  ay,  the  temperature  being 
fixed  from  the  equation  pv  =  RT.  If 
now  the  pressure  of  the  gas  is  increased 
from  ay  to  by,  there  being  no  change 
in  volume,  there  will  be  network  done  , 

A  /* 

by  the  air.  As  its  pressure  is  increased 
the  temperature  is  increased  in  the  same  proportion, 
and  we  must  have  added  enough  heat  to  cause  this 
change  in  temperature.  If,  however,  instead  of  in- 
creasing the  pressure,  it  had  been  maintained  constant 
and  the  volume  increased  from  oy  to  ox,  we  would 
have  had  not  only  to  raise  the  temperature,  but  to 
have  done  work  overcoming  a  pressure  ay  through  a 
distance  yx. 

Again,  if  neither  the  pressure  nor  the  volume  re- 
mains constant,  we  have  in  Fig.  2  the  condition  a  for 
the  initial  condition  and  d  for  the 
final  condition,  and  the  amount  of 
heat  which  must  have  been  added 
from  a  to  d  must  have  been  enough 
to  change  the  temperature  from 
that  at  a  to  that  at  d,  and  also  to 
do  an  amount  of  work  equal  to  the 
area  ayxd. 

To   express  the  relation   between   these    quantities 
we  must  have  units  in  which  to  measure  them. 

The  unit  in  which  the  quantity  of  heat  is  measured 
is   the   amount    of   heat   which    must   be   added    to    I 


-  • 


y    x 
FIG.  2. 


NO  TES  ON  T HER  MOD  YNA  MICS.         7 

pound  of  water  at  62  degrees  F.  to  raise  its  tempera- 
ture to  63  degrees,  and  is  called  the  British  Thermal 
Unit,  or  simply  B.T.U. 

The  unit  of  work  is  the  foot-pound,  arid,  experi- 
mentally, it  has  been  determined  that  one  B.T.U.  is 
equivalent  to  778  foot-pounds. 

The  number  of  heat-units  which  must  be  added  to 
I  pound  of  any  substance  to  raise  it  I  degree  in  tem- 
perature is  called  the  specific  heat. 

Referring  now  to  Fig.  i,  if  cv  is  the  amount  of 
heat  which  must  be  added  per  degree  to  raise  the 
temperature  from  a  to  b,  then  cv  is  the  specific  heat 
for  constant  volume,  and  the  total  heat  required  is 
cv(Tb  —  Ta)  when  Tb  and  Ta  are  the  temperatures  cor- 
responding to  the  conditions  b  and  a  respectively. 
The  value  cv  for  air  is  .169  heat-unit,  or  132  foot- 
pounds. 

Prob.  10, — If  5  cubic  feet  of  air  at  30  pounds  pressure  per 
square  inch  and  60  degrees  F.  has  20  heat-units 
added  to  it  at  constant  volume,  and  if  the  heat 
required  to  raise  the  temperature  of  i  pound  t 
degree  at  constant  volume  is  .169  heat-unit,  what, 
is  the  resulting  temperature  ? 

The  weight   of  air  is  —  ^  =  .774  pound.     The 

53-37X52I 

heat  required  to  raise  this  i  degree  is  . 774  x. 169 
=  .131   heat-unit.      The   rise   in    temperature   is 

»  therefore  —  -  =153  degrees. 

Prob.  11.— If  15  cubic  feet  of  air  at  100  pounds  pressure  per 
square  incii  is  raised  from  60  degrees  to  100  de- 
grees F.  at  constant  volume,  how  much  heat  is 
inquired?  5"  Z  <J 


8  NOTES   O.V    THERMODYNAMICS. 

Similarly,  if  cp  is  the  total  amount  of  heat  pet- 
degree  which  must  be  added  to  the  pound  of  gas  at 
a,  Fig.  I,  to  cause  the  gas  to  expand  from  a  to  c, 
then  cp  is  the  specific  heat  for  constant  pressure, 
and  the  total  heat  required  is  cp(Tc  —  Ta).  In  this 
case,  however,  the  heat  has  been  used  partly  in  rais- 
ing the  temperature,  the  remainder  being  required  to 
do  external  work.  We  can  therefore  write  for  the 
quantity  required  to  change  the  temperature  only, 
cv(Tc  —  Ta),  and  for  the  quantity  required  to  do  the 
work,  ay(ox  —  oy)  or  fa(vc  —  va)>  and,  as  all  the  heat 
must  be  accounted  for,  we  can  write 


For  air  cp  is  .238  heat-unit*  or  185  foot-pounds. 

Prob.  12. — If  i  pound  of  air  is  changed  from  20  degrees  to  30 
.  degrees  F.  at  a  constant  pressure  of  100  pounds 
per  square  inch,  how  much  heat  must  be  added  if 
to  raise  the  temperature  alone  required  that  the 
equivalent  of  132  foot-pounds  of  work  be  added 
for  each  degree  ? 

The  heat  to  change  the  temperature  only  is  the 
equivalent  of  (30  —  20)  x  132  =  1320  foot-pounds. 
The  amount  of  work  to  be  done  is  to  overcome 
the  pressure  of  100x144  pounds  per  square  foot 
through  the  difference  in  volume.  The  initial 

volume   is   ^^ =  1.79   and   the   final   vol- 

100 x 144 

ume  =  •— — - — ^— =  1.8^.     The  work  is   then    100 
100 x 144 

x  144(1.83  —  1.79)  =  576.0  foot-pounds.  The  total 
heat  required  is  therefore  the  equivalent  of  1320 
+  576  =  1896  foot-pounds. 


NOTES   ON    THERMODYNAMICS.  9 

Prob.  13.  — If  cv  =  132  foot-pounds,  prove,  by  using  equation  (2), 
that  cp=  185  foot-pounds.  j£  t    $ 

Taking  now  the  third  case,  if  we  call  cn  the  total 
amount  of  heat  per  degree  which  must  be  supplied 
from  a  to  d,  Fig.  2,  then  cn  is  the  specific  heat  for  the 
law  represented  in  the  figure.  This  is  used  up  partly 
in  changing  the  temperature,  which  will  account  for 
the  amount  cv(Td—  Ta),  and  the  balance  in  doing  the 
work  represented  by  the  area  ayxd.  We  can  there- 
fore write 

cn(Td-Ta}  =  cv(Td~Ta}  +  ay*d.    .    .    (3) 

The  two  equations  above  can  be  written  in  the  gen- 
eral form, 

Total  Heat  = 

Heat   required   to  raise   temperature  -J-  work  done, 

or,  in  the  differential  form, 

dH  —  cvdt  +pdv,      ....      (4) 


the  latter  term  being  the  calculus  method  of  indicating 
the  elementary  area  ay'x'd'. 

This  equation  is  the  fundamental  one  of  the  thermo- 
dynamics of  gases. 

Equation  (2)  can  be  written  as  below  from  the  fact 
that  pcvc  -RTC,  and  pava  =  R Ta : 

/•  (  T        T  \  /-  (  T        T\  _L  E>(  T        T\ 

Cp\l  c —  1  a)   —  Cv\  1  c —  1  a)  -{-  /v^  1  c —  1  a), 

or 


1 0  NO  TES   ON    T HER  MOD  YNAMICS. 

This   equation   represents   the    relation    between    the 
quantities  which  it  is  important  to  remember. 

Experimentally,  it  has  been  shown  that,  for  perfect 
gases, 

^  =1.41, 

cv 

and  we  can  write 

cp\cv\  R  ::  1.41  :  i  :  .41, 
or 

Heat   added   at    constant   pressure  :  Heat  required  to 
raise  the  temperature  :  the  work  done  :  :  1.41  :  I  :  .41. 

Prob.  14. — If  5  pounds  of  air  at  170  degrees  F.  has  16  heat-units 
added  to  it  at  constant  pressure,  how  much  work 
is  done  ?     What  is  the  final  temperature  ? 
To  find  the  work  done  we  have  5  x  c .(ti  —  170) 


=  i6h.  u. 


Work  =  5  x  R(t*  —  170)  =  —  x  16  x  778  ft.-lbs. 

=  3620  ft.-lbs. 

Work  3620 

-The  rise  in  temperature  =  —    -  =  — - —    —  =  13  .6. 

5^          5  x  53-37 

The  final  temperature  is  170  +  13.6  =  183.6. 

Prob.  .15. — A  given  weight  of  air  expanding  at  constant  pressure 
does  1000  foot-pounds  of  work.  What  heat  must 
have  been  added  to  the  air  ?  How  much  heat  was 
used  to  raise  the  temperature? 

Prob.  16. — 15  cubic  feet  of  air  expands  to  40  cubic  feet  under  a 
constant  pressure  of  30  pounds  per  square  inch. 
How  much  heat  was  required  ? 

Now/z>  —  RT,  and  if  all  are  variables,  we  can  write 
pdv  +  vdp  =  Rdt, 


NOTES   ON    THERMODYNAMICS.  II 

and  substituting  this  value  of  pdv  in  (4),  we  have 

dH  =  cvdt  +  Rdt  —  vdp, 
or,  from  (5), 

dH  =  cpdt  -  vdp (6) 

The  two  equations  (4)  and  (6)  are  often  spoken  of  as 
the  two  fundamental  equations  of  the  thermodynamics 
of  perfect  gases. 

The  quantity  of  heat  required  to  cause  I  pound  of 
air  to  expand  doing  work  can  then  be  written  as 
follows : 


\  pdv,      .     .      . 
J  *i 


H=c,(T,-  r,)+  I  pdv,     ...     (7) 

fc/P, 

in  which  T2  is  the  final  temperature,  v2  the  final  vol- 
ume, and  7\  and  z/t  the  corresponding  initial  condi- 
tions. 

Prob.  17.  —  If  the  initial  condition  is  such  that  5  pounds  of  air 
occupy  50  cubic  feet  at  30  degrees  F.,  and  the  final 
condition  such  that  it  occupies  120  cubic  feet  at  40 
degrees  F.,  and  the  expansion  takes  place  along  a 
straight  line,  how  much  work  is  done  and  how 
much  heat  added  ? 
It  is  first  necessary  to  find  the  pressure.  From  pv=RT 


we  have  for  the  initial  state,  p  =      '  -   =  2630 


pounds  per  square  foot.     For  the  final  condition, 
p  _  53-37         _  _  II2Q   p0uncis   per  SqUare  foot. 

~5~ 

The  work  done  is  therefore,  from  a  diagram, 
2630+  ii2oi2o_    Q  _      jooot-ies  h.  u. 

' 

OF 

:?••• 


12 


NOTES   ON    THERMODYNAMICS. 

The  heat  required  to  raise  the  temperature  is 


and  the  total  heat  required  is 

1 68  4-  8  =  176. 

To  determine  the  value  of  area  abed  or  the 


I  pdv, 


we  must  know  the  law  connecting 
3pvT2  the  pressure  and  the  volume  of 
the  path  ab.  If  we  call  this  pvn 
=  K,  we  have,  knowing  pv  vl  and 
71,  and/2,  v2  and  7"2 , 


d  c 

FIG.  3. 


or 


n  = 


log  A-  lQg  A 


The  value  of  K  is  obtained  from  either  of  the  above 
equations. 

Prob.  18^  —  What  is  the  value  of  n  that  the  expansion  curve 
passing  through  the  same  initial  and  final  points 
as  in  problem  (17)  should  be/z/M  =  K1 


Area  of 


Tkdv      K 


••••  dv      K  r      i  i*. 

*j*  =        ^  =  —  [- 


n  — 


NOTES   ON    THERMODYNAMICS.  13 

Putting  in  the  value  of  K  for  the  equation  above,  we 
have  the  work 


w= 


I  —  n 


Prob.  19.—  Having  given,  in  problem  (17),  that  the  law  of  the 
expansion  is  /z/-975  =  K,  how  much  work  is  done 
if  the  final  condition  is  /  =  40°  ? 

Work  =  5  x     ^3'37    (40  —  30)  =  107000  ft.-lbs. 
The  total  quantity  of  heat  required  is  therefore 

cv(T2-Tl)  +  Tlt(T2-Tl\     .     .     .     (10) 
cv-nc.+R  ,_cp-nc« 


l_n 


when   CH    is   the   specific   heat    according    to    the    law 
pv"  =  K. 

Equation  (10)  is  worth  committing  to  memory  as  it 
is  here  given. 

Prob,  20,—  How  much  heat  would  be  required  in  problem  (19)? 

From  problem  (17)  the  heat  required  to  change  the 

temperature  is  8  heat-units.     From  (19)  the  work 

done  =  I07°800  =  137  heat-units.     The  total  heat 
required  is  137  4-  8  =  145  heat-units. 


14  NOTES   ON    THERMODYNAMICS. 

The  special  cases  already  treated  of  and  some  others 
may  readily  be  derived  from  equations  (9)  and  (10). 

In  n  =  o,  pvn  —  K  becomes/  —  constant,  the  work 
done,  frpm  (9),  is,  evidently,  R(T2  —  7^),  and  the  heat 
required,  from  (10),  is  (cv  +  R)(T2  —  TJ=  cP(T2  —  TJ 
as  before. 

If  —  =  o,  we  have  v  =  constant,  and  the  work  done, 
n 

from  (9),  is  evidently  o,  as 


The  heat  required,  from  (10),  is  cv(  T2  —  T^. 

If  the  heat  is  constant,  we  have  -         --(T2—  7'1)=o, 

and   one  solution   of  this  is  —  =  n.     This  expansion, 

where  no  heat  is  added   nor  taken  away  but  work  is 
done,  is  called  adiabatic  expansion,  and  its  equation  is 

pvcv=  Ky  or,  as  for  air  --  =  1.41,  we  have 

Cv 

pvlAl  =  K. (n) 

The  work  done  is 

R     (T         T\        —(T         T^ 

*-n(    2~"      1>-7^~1(    l"      2)* 


NOTES   ON    THERMODYNAMICS.  I  5 

Evidently,  as  cp  ~  R  +  cv,  and  cp  =  1.41^,  we  have 
R  =  -4it\,,  and  the  work  done  is,  for  adiabatic  expan- 
sion, cv(T2  —  7\),  and  the  heat  given  up  is  cv(T2  —  7^) 
to  do  this  work. 

If  the  temperature  is  kept  constant  we  have  Tl~T2J 
pv  =  RT  =  K,  and  n  =  I.  The  amount  of  heat  re- 
quired is  then,  from  equation  (9), 

<±^(T      r)^*, 

I    —    I  V     2  O' 

which  is  indeterminate.  We  can,  however,  determine 
the  quantity  of  work  done  and  of  heat  added  by  going 
back  to  the  original  equation, 


-   73.+ 


Here  Tz  =  Tv  and  pv  —  K.      Consequently 

crW 

e^      (12) 
V\ 

Evidently,  from  the  equation  pv  =  RT,  we  can  put 
^  for  either/^  or/22/2,  and,  from  /z/  =  7T,  we  can 


put  for  —  the  value  —  .    In  solving  problems,  that  form 

v\  A 

of  equation  should  be   used  which   covers  the  greatest 

amount  of  given  data. 

Prob,  21.—  If  i  pound  of  air  has  40  heat-units  added  to  it  and 
25  heat-units  are  the  equivalent  of  the  external 
work,  what  is  the  value  of  n  in  the  equation 


1 6  NO7^ES   ON   THERMODYNAMICS. 

As  the  external  work  =  25  h.  u.  =  (7^3  —  Ti),  the 

I    ™~*    ft 

remainder,  15  h.  u.,  =cv(Ti  —  7^),  or 

11  =  Cv^1  ~  n)  —  l  ~  H 
25  ~          R  .41 

«  =  .754. 

Prob.  22. — If  10  heat-units  are  added  to  i  pound  of  air  at  con- 
stant pressure,  what  work  is  done  and  what  is  the 
rise  in  temperature  ?  We  have 

cp\cv\  7v'  : :  Heat  added  :  Heat  to  raise  tempera- 
ture :  work 

10       10  x.4i 
: :  1.41  :  r  :  .41  : :  10  :  —   :  — 

1.41         1.41 

10  x   .41   x  778 

Work  =  -  -f-L.  =  2270  ft.-lbs. 

1.41 

As  we  are  dealing  with  i  pound,  the  rise  in  temper- 
ature 

Work 

.,  -  =  42.4  degrees. 

Prob.  23.— If  40  heat-units  are  added  to  5  pounds  of  air  having 
a  pressure  of  25  pounds  per  square  inch  and  a 
volume  of  30  cubic  feet,  what  is :  (i)  final  v,  /,  / ; 
(2)  the  work  done  if  (A)  it  is  added  at  constant 
pressure,  (B)  at  constant  volume,  (C)  at  constant 
temperature,  (D)  according  to  the  law/z/*  =  A"? 

The  value  of  ;/  when  no  heat  is  added  could  have 
been  determined  directly  from  the  fundamental  equa- 
tions as  follows :  When  no  heat  is  added  we  can 
write 

dH  =  c^t  -f-  pdv  —  o,      or     cvdt  =  —  pdv, 
and 

dH  —  Cpdt  —  vdp  =  o,      or     cpdt  —  vdp, 


NOTES   ON   THERMODYNAMICS. 


and  dividing  one  by  the  other  we  have 

cv          pdv  c,,   dp  dv 


CP 


vdp' 


or     —  .  —  —  — 


or  integrating  between  limits  we  have 


or,  dropping  the  logarithms, 


or 


or 


To  determine  whether  the  temperature  will  rise  or 
fall  during  expansion,  whether  work  must  be  done  by 
the  air  or  on  the  air,  and  whether  heat  must  be  added 
or  taken  away,  Fig.  4  will  be  of  service.  Through 


£•0 


n=o 


FIG.  4. 

the  initial  point  A,  Fig.   4,  we  have  drawn  a  series 
of   curves    for    different    values    of    n.     n  —  o    is    at 

constant  pressure,  —  =  o  is  at  constant  volume,  n  =  I 

is  at  constant  temperature,  and  n  =    1.41   is  an  adia- 
batic. 

Evidently  all  expansion  curves  having  n  positive  will 


1 8  NOTES   ON    THERMODYNAMICS. 

fall  between  a  and  d,  all  having  n  negative  will  fall 
between  a  and  //.  All  compression  curves  having  n 
positive  will  fall  between  e  and  h,  and  negative  values 
will  fall  between  d  and  e. 

Starting  at  A,  if  the  path  of  the  air  is  to  the  right, 
work  is  done  by  the  air,  or  is  positive;  if  to  the  left, 
work  is  done  on  the  air,  or  is  negative.  The  following 
table  should  be  mastered  by  the  student.  From  A, 
then,  calling  rise  in  temperature,  heat  added,  or  work 
done  by  the  air  positive,  we  have,  if  curve  falls  between 
the  limits, 

n.  Temp.         Heat.         Work. 

a  to  b    o  <  n  <  I  +  +  -j- 

b  to  c I    <  n  <   1. 4 1  -j-  -|- 

c  to  d 1.41  <  n  + 

d\.o  e n  <  o 

e  to/    o  <  n  <  I 

ftog I   <  n  <  1.41        + 

gto  h 1.41  <  n  +  + 

hto  a n  <  o  +  +  + 

We  are  now  ready  to  take  up  the  question  of  the 
amount  of  heat  expended  and  the  amount  of  work 
done  when  the  gas  under  consideration  goes  through 
a  series  or  cycle  of  changes,  being 
at  the  end  in  the  same  condition  as 
at  the  beginning.  In  Fig.  5,  sup- 
pose we  have  a  pound  of  the  gas 
starting  at  the  condition  p£\Tl  and 
~  expanding  according  to  the  law 

pvm  ==    K  until  it    reaches  a    point 
Suppose    now    it    expands    along    the    line 


NOTES   ON   THERMODYNAMICS.  1$ 

pv"  =  K^  to  the  condition  psvsT3.  It  is  then  com- 
pressed along  the  \me  pvm  =  K^  to  p^v^T^  ,  which  is 
such  a  point  that,  if  the  compression  is  continued 
along  the  line/z/1  =  K%  ,  it  will  again  reach  its  initial 
condition. 

There  are  certain  algebraic  relations  between  the 
quantities  in  this  diagram  which  should  first  be  de- 
duced. They  are  : 

A_A.        ^_^3.        j^2_    ^3 

A~A  '    ^i~V     T,~''  TV 
From  the  given  data  we  have 


and  multiplying  tfe^ese  equations  together  we  have 


V-  Z>3  /        N 

?-£      •     • <'3) 

Again, 

A 
A  = 

or 

rr  =  f^  =  x> from  Os); 


A=A       .....     (I4) 

Pi     A 

Multiplying  (13)  by  (14)  we  have 

^  =  g  =  .^=^  _  _  >  (I5) 

These  relations  should    be  kept  in   mind,   as  they 
often  lead  to  an  easy  solution  of  problems.     Equations 


2O  NOTES   ON   THERMODYNAMICS. 

(13)  and  (14)  are  true  if  the  figure  is  bounded  by  any 
two  similar  (algebraic)  sets  of  curves,  and  equation  (15) 
is  only  true  for  substances  having/^  =:  RT  for  their 
equations. 

The  work  done  in  a  cycle  similar  to  that  in  the 
figure  is  evidently  the  area  of  the  diagram,  or  it  is  the 
heat  added  from  4  to  2,  less  that  taken  away  from  2 
to  4. 

From  4  to  i, 


Tt);      Work=(r1-  Tt). 


From  I  to  2, 


From  2  to  3, 

Heat  =  (<:„+  ^)  (  7-3-  7y  ;       Work^y-^  Ts- 
From  3  to  4, 


The   net  work    done   is  therefore  the   area   of  the 
diagram,  or 

-  Tt-  Tt)  +  -^-(-  T-  Ts+ 


i  — wv  i  —  m 


The  total  quantity  of  heat  which  must  be  added  is 


NOTES  ON    THERMODYNAMICS.  21 

that  required  to  raise  the  body  from  Tt  to  T2  through 
T     or 


=  total  heat  added  ;   or,  calling  cm  and  rw  the  specific 
heats  according  to  the  laws  I,  2  and  4,  I,  we  have 

Total  heat  =  cn(T,  -  Tt)  +  cm(Tt  -  T,). 

The  efficiency,  which  is  the  ratio  of  the  work  done 
to  the  heat  expended,  is  then 


If  either  set  of  curves  is  adiabatic  we  have,  say  for 
n  =  1.41,  for  the  efficiency 


As  R  =  .41^  ,  we  have 


o 

^  -Tt 

T,  -  r; 


~ 


T  T 
Putting  T3  =  -^  ,  we  have  for  the  efficiency 


06) 


22  NOTES   ON   THERMODYNAMICS. 

That  is,  in  any  such  cycle,  the  efficiency  is  the  drop 
in  temperature  along  either  adiabatic  divided  by  the 
highest  temperature  on  that  adiabatic.  The  ammint 
of  work  done  in  such  a  cycle  can  be  determined  by  mul- 
tiplying the  heat  added  by  this  efficiency. 

Prob.  24. — A  cycle  is  made  up  of  two  adiabatics  and  two 
curves  pw*  =  K.  If  10  heat-units  are  added  to 
i  pound  of  air,  /i  =  3000  pounds  per  square 
foot,  Vi  =  10  cubic  feet,  how  much  work  will  be 
done,  the  lowest  temperature  in  the  cycle  being 
o  degrees  F.,  and  what  is  the  highest  tempera- 
ture in  the  cycle  ? 
In  Fig.  6  we  have  the  data  given  as  shown.  To 

3000  x  10 
determine    T \ ,  we   have    7\  =  —  -  =  561. 

53-37 
The  work  done  is 

561  —  461 
10  x  - —  x  778  =  1390  ft.-lbs. 

To   determine   7\,  we  know  that  10  heat-units   are 

added  from  T\  to  7"2  ac- 
cording to  the  law  pv^ 
=  K,  or 


IG  = 


7781.41 


7^-7^  =  32.8,     T,  =  593-8. 

Prob.  25.— A  cycle  is  made  up  of  two  isothermals  and  two  con- 
stant-volume lines.  The  extreme  volumes  are  40 
and  10  cubic  feet,  and  the  extreme  pressures  are 
15  and  100  pounds  per  square  inch.  How  much 
work  is  done  and  how  much  heat  is  required  ? 

Prob.  26. — A  cycle  is  made  up  of  two  constant-pressure  and 
two  isothermal  lines.  The  extreme  pressures  are 


NOTES   ON    THERMODYNAMICS.  2$ 

15  and  10  pounds  per  square  inch,  and  the  ex- 
treme volumes  are  10  and  70  cubic  feet.  How 
much  work  is  done  and  how  much  heat  is  re- 
quired ? 

Prob,  27.  —  Having  given  2  pounds  of  air  at  /i  —  3000  pounds, 
z/i=  15  cubic  feet,  7^=460,  7"3  =  420,  and/z/-7=A", 
how  much  work  is  done,  the  other  curves  being 
adiabatics  ? 

In  a  cycle  such  as  we  have  just  been  considering  it 
can  be   shown  that  the  work   done 
may   be   expressed  in   a   number  of 
ways.      In    Fig.    7   the    heat   added 
from    T,   to    T2  =  cH(T2  -  T,)  =  Qr 

The  heat  taken  away  from  T9  to  Tt 

FIG    7 

=  c.(Tt-  Tt)=Q2.     The   first   of 

these  divided  by    7^  is  equal  to  the  second  divided 

by  7'4.      For  we  have  the  relation 

T        T 

IJ—  _3 

T  ~"  T  ' 

f  1  ^4 

and,  therefore, 


f.(Tt- 


T  T 

2  I  2  4 

In  the  same  way  the  heat  along  the  top  line  di- 
videcl  by  T2  is  equal  to  the  heat  along  the  bottom 
line  divided  by  Ty  The  work  in  such  a  cycle  can 
therefore  be  stated  as  the  heat  added  along  either 
line  divided  by  the  temperature  at  either  end  of  the 
line  taken  and  multiplied  by  the  range  oi  tempera- 


24  NOTES   OK    THERMODYNAMICS. 

ture  along  the  adiabatic  passing  through  the  point  at 
which  the  temperature  was  taken,  or  the  work  is 

||(7\  -Tt)  =  Q  (T*  -  TV)  =  %(T,  -  TV) 

/I  A  74 


=  ?  (TV  -TV).  (.7) 


3 


This  relationship  should  be  entirely  understood. 

Having  shown  that  the  work  done  in  any  cycle 
having  adiabatic  curves  for  two  of  the  bounding 
curves  is  equal  to  the  heat  added  times  the  range  in 
temperature  along  one  adiabatic  divided  by  the  max- 
imum temperature  along  that  adiabatic,  it  can  be 
shown  that,  if  the  heat  is  added  at  constant  tempera- 
ture, the  maximum  range  in  the 
cycle  being  the  same,  the  amount  of 
work  done  or  the  efficiency  will  be 
the  greatest.  In  Fig.  8  let  1,2  and 
FlG  g  3,  4  be  isothermals,  and  2,  3  and 

I,  4  be  adiabatics.     Then  I  and  2  will 
be   at  the  highest  temperature,  and   3  and  4   at  the 

'p  j" 

lowest.     The  efficiency  is  then  -^-= — 4.      Now,  sup- 

f\ 

pose  the  heat,  instead  of  being  added  along  an  iso- 
thermal, is  added  according  to  any  law  as  1,5.  The 
temperature  of  5  is  evidently  below  I  or  2,  as  we  have 
assumed  that  7i  is  the  highest  obtainable  temperature. 
As  7"3  is  the  lowest  temperature,  it  is  evident  that  a 
curve  similar  to  I,  5  passing  through  3  will  cut  I,  4  at 
a  higher  temperature  than  7"4.  The  efficiency  is  then 


NOTES    ON   THERMODYNAMICS. 


IS 


y  _  y  y  y 

^-~, — 6,  which  is  evidently  less  than  -^--TR — 4,  as  7"6  i 
1 1  *i 

greater  than  7^.  As  the  efficiency  is  less,  the  work 
done  by  the  same  quantity  of  heat  is  less.  Therefore 
the  greatest  efficiency  is  obtained  when  heat  is  added 
at  constant  temperature,  which  also  implies  that  heat 
must  be  taken  away  at  constant  temperature. 

The  diagrams  we  have  drawn  heretofore  have  shown 
the  amount  of  work  done,  but  have  given  us  no 
graphical  idea  of  the  quantity  of  heat  which  enters  the 
cycle.  This  quantity  of  heat,  as  well  as  the  quantity 
of  work,  can  be  shown  by  a  definite  area  on  this 
diagram.  By  a  definite  area  is  meant  one  that  can  be 
measured  by  a  planimeter. 

Suppose  1,2,  Fig.  9,  to  be  the  path  representing 
the  changes  in  pressure  and  volume.  We  have  the 


10    11 


FIG.  9. 


work   done   A  -f-  B  -f-  C,    the   letters   referring   to  the 
spaces  in  which  they  occur. 

AL  i  the  total  energy  in  the  gas  can  be  represented 
by  drawing  the  adiabatic  I,  12,  6  and  continuing  it 
indefinitely  to  the  right.  The  area  under  this  curve, 
or  C  -\-  F -{-  G,  is  the  equivalent  of  the  energy  in  the 
substance  at  i.  because  it  is  the  amount  of  work 


26  NOTES   ON    THERMODYNAMICS, 

which  would  be  done  if  it  was  allowed  to  expand  at 
the  expense  of  its  own  heat  until  it  reached  the  abso- 
lute zero.  At  2  the  energy  remaining  in  the  gas  can 
be  represented  by  the  total  area  under  the  adiabatic  2,  3 
drawn  through  2.  This  is  equal  to  D-\-E-\-F-\-H-\-G. 
We  have  then  that  the  amount  of  heat  added  is  equal 
to  the  energy  remaining  at  2  plus  the  work  done  from 
I  to  2  and  minus  the  energy  at  i,  or 

Heat  added 


or  the  area  between  the  path  i,  2  and  two  adiabatics 
drawn  through  the  extremities  of  the  path  and  indefi- 
nitely extended. 

We  have  already  seen  that  the  work  done  by  a 
pound  of  air  expanding  adiabatically  can  be  repre- 
sented by 


where  T2  is  the  final  and  7i  is  the  initial  temperature. 
The  energy  in  a  pound  of  gas  at  I  can  be  determined 

TT)  rri 

"  by  making  T2  in  the  above  equation  o,  and  -  -  or  — 
is  the  energy.      Similarly  at  2  the  energy  in  a  pound 

of  the  gas  is  -  .      If  2,  4  is  an  isothermal  through 
.41 

2,  the  energy  in  the  gas  at  2   is  the  same  as  at  4,  or 


NOTES   ON    THERMODYNAMICS.  2f 

RT 

—  ,  and  if   I,  3,  5   is-  an   isothermal  through    I,  the 

.41 

energy  in  the  gas  at  I,  3,  or  5  is 


.41        .41 

Evidently,  if,  after  expansion  takes  place  from  I  to 
2,  we  allow  it  to  continue  adiabatically  to  3,  the  air 
has  as  much  energy  at  3  as  it  had  at  I,  and  whatever 
heat  we  have  added  has  all  gone  to  do  work.  The 
total  work  done  is  (A  +  B  +  C  +  D  +  E  +  F),  and 
this  is  equal  to  the  heat  added  from  I  to  2. 

The  area  D  +  E  +  F  is  equal  to  the  area  K  -\-  L, 

7?  T* 

for  at  2  the  energy  in  the  gas  is  --  2,  and  at  4  it  is  the 

.41 

r>  y 

same.      At  3  the  energy  is  --  -,  and  at  5  it  is  the  same. 

.41 

Passing  from  2  to  3  the  energy  converted  into  work  is 

r> 

—  (Tz—  7\),  and  from  4  to  5  it  is  the  same.    But  the 

work  done  is  in  one  case  D  -\-E-\-  F,  and  in  the  other 
K-\-L\  and  as  they  are  the  equivalent  of  the  same 
amount  of  energy,  they  are  equal  to  each  other. 

Prob.  28.  —  How  much  energy  is  there  in  I  pound  of  air  after  it 
has  expanded  adiabatically  to  20  cubic  feet,  if  its 
initial  conditions  were/  =  2000  pounds,  v  =  16 
cubic  feet  ? 

Prob.  29.  —  What  is  the  energy  in  10  cubic  feet  of  oxygen  at 
IGU  pounds  pressure  per  square  inch  and  100 
degrees  F.  ? 


28 


NOTES   ON    THERMODYNAMICS. 


There  is  another  method  of  illustrating  graphically 
the  heat  added  under  any  conditions.  If  we  attempt 
to  draw  a  diagram  having  absolute  temperature  T  for 
ordinatesand  Q,  the  heat,  for  the  area  under  any  curve 

- 


to  the  other  axis  of  co-ordinates,  the  abscissa  is 


because  Q  = 


The  quantity     /  -£ 


called  _enr< 

Evidently  on  such  a  diagram  an  adiabatic  is  repre- 
sented by  a  line  parallel  to  the  T  axis,  because  no 
heat  is  .added  along  an  adiabatic.  The  diagrams 
shown  in  Figs.  10  and  n  represent  a-/,  v  diagram 


and  a 


3000- 


10  go 

VOLUME  IN  CUBIC  FEET. 

FIG.  10. 


The  data  assumed  in  drawing  these  diagrams  are 
PA  —  3000>  TA  =  561,  VA=  10,  VB  =  20;  for  AB, 
n  =  o;  AC,  n  —  i  ;  AD,  n  =  1.41  ;  CEy  n  —  1.41; 
and  for  DE,  n  =  i. 


NOTES   ON   THERMODYNAMICS.  2$ 

In  locating  points  in  Fig.  n,  the  point  A  is  taken 
at  any  point  on  the  T  =  561°  line.  To  determine  the 
distance  to  C,  we  have,  as  this  is  a  constant-tempera- 
ture line,  dQ  =  pdv,  and 


f  -/?-*/*  =—5=3;., 

To  locate  the  point  B,  we  have  dQ  —  cpdt  and 


These  diagrams  are  drawn  to  such  a  scale  that  the 
area  represents  foot-pounds  in  either  diagram.  In  the 
first  diagram,  Fig.  10,  the  area  under  AB  is  the  work 
done  at  constant  pressure,  and  in  the  second  diagram, 

1.41 
Fig.  u,  it  is  the  heat  added  and  is  -  as  great.    In 

_        ,    -  --  _     i        ._  _    ,         -  -  __   '~  .....  -r4>  t  -  .|T<  1M.rr|.  mfm  _  ___^p»» 

the  first  the  area  under  AC  is  the  work  done  at  con- 
stant temperature,  and  in  the  second  it  is  the  heat 
added  and  is  exactly  equal  to  it.  In  the  first  the 
area  under  AD  is  the  work  done  adiabatically,  and  in 
the  second  it  is  zero,  as  it  should  be. 

If  we  draw  through  D  an  isothermal  as  shown  by 
the  line  DE,  the  point  E  completes  a  cycle,  and  for 

the    second    figure    evidently    —^  —  ^r^>    as    proved 

•1  AC  ^  DE 

above,  and  the  areas  AC  ED  in  the  two  figures  are 
equal. 

Prob.  30.  —  Draw  diagrams,   similar   to    Figs.    10   and    n,   to 
scale  representing  the  expansion  of  i  pound  of  air 


3O  NOTES   ON   THERMODYNAMICS. 

at  60  pounds  pressure  and  100  degrees  F.  (A) 
adiabatically,  (B)  along  the  isothermal,  (C)  at  con- 
stant pressure,  until  the  volume  is  doubled,  and  in 
each  case,  if  possible,  represent  by  a  definite  area 
the  amount  of  work  done  and  energy  expended. 


GENERAL     EQUATIONS. 

In  taking  up  the  portions  of  thermodynamics  treat- 
ing of  substances  generally,  certain  matters  which  we 
have  already  deduced  apply,  while  certain  others  do 
not.  Thus,  Fig.  12,  if  AB\s  the  path  of  the  substance 
under  discussion  (any  substance),  the 
external  work  done  is  here,  as  before, 
*.  the  area  ABDC.  The  total  amount 

B- 

of  head  added  to  cause  the  substance 
to  pass  from  A  to  B  is  again  repre- 
sented by  the  area  between  AB  and 
°  PT^DTO       F    two  adiabatics  at  the  extremities  A 

I  1G.    12. 

and  B  indefinitely  extended  to  the 
right.  Here,  however,  the  adiabatics  are  not  neces- 
sarily curves  whose  equation  is  pvlAl  =  K,  as  this  rela- 
tion only  applies  to  perfect  gases.  They  are  curves, 
however,  so  drawn  that  from  B  to  E,  for  instance,  the 
area  BEFD,  which  is  the  external  work  done,  is  the 
exact  equivalent  of  the  heat-energy  which  has  disap- 
peared as  such  between  B  and  E. 

We  have  called  certain  lines  isothermals,  and  made 
certain  statements  about  these  lines.  That  is,  in  Fig. 
13,  if  AB  is  an  isothermal  fora  perfect  gas,  it  is  a  rect- 
angular hyperbola,  the  heat  added  from  A  to  B  is 
the  area  L'BAL  and  is  exactly  equal  to  the  area 


NOTES   ON    THERMODYNAMICS. 


c 
FIG.  13. 


A  BCD  representing  the  external  work.  Hereafter 
AB,  if  it  is  an  isothermal,  is 
only  a  line  of  constant  tem- 
perature; it  need  not  be  and 
often  is  not  a  rectangular  hy- 
perbola. The  heat  added  is 
equal  to  L'BAL  but  is  not 
necessarily  equal  to  ABCD. 
The  work  done  is  equal  to  ABCD  and  may  or  may 
not  be  equal  to  L'BAL. 

The  attempt  will  be  made  hereafter  to  use  the  terms 
adiabatic  and  isothermal  in  the  general  sense  spoken 
of  above. 

Fig.  14  shows  the  work  done,  and  Fig.  15  the  heat 
added  isothermally  to  any  substance.  In  Fig.  14  the 


VOLUME 

FIG.  14. 


ENTROPY 

FIG.  15. 


isothermal  may  be  a  rectangular  hyperbola  if  we  are 
dealing  with  air,  a  constant-pressure  line  if  we  are  deal- 
ing with  a  mixture  of  liquid  and  vapor,  or  it  is  the  line 
which  represents  the  relation  between  /  and  v  at  con- 


32  NOTES   ON   THERMODYNAMICS. 

stant  temperature.  In  Fig.  15  it  must  be  a  line  perpen- 
dicular to  the  T axis.  ALand  B'  areadiabatics;  in  Fig. 
14  they  are  curves,  and  in  Fig.  1 5  they  must  be  straight 
lines  parallel  to  the  Taxis.  The  heat  //added  from  A  to 
B  in  both  diagrams  is  the  area  ABL'L.  Draw  any  other 
isothermal  A' B'  in  both  diagrams  so  that  its  tempera- 
ture is  dt  degrees  below  AB.  Evidently,  from  Fig.  1 5, 

JT 

the  area  ABB' A'  is  equal  to  -^dt.    From  Fig.  14,  the 

equal  area.  ABB' A'  is  /  dp  dv,  and  these  two  quantities 
are  equal  to  each  other,  or 


H  i 

—  dt  —    I  dpdv,      .     .     .      .     (18) 


where  dp  is  the  vertical  distance  between  AB  and 
A'B1 ' ,  or  dv  is  the  horizontal  distance  between  these 
lines,  but  not.  both  at  the  same  time.  We  can  write 
the  equation  in  either  of  the  following  forms: 

H  CVB  CPB 

Ydt  =      /    (dp)  dv  —    \    (dv)dp, 

the  quantity  in  the  parenthesis  meaning  that  the  value 
of  (dp]  is  fixed  by  the  isothermals  and  that  dv  is  the 
other  independent  variable,  or  in  the  last  member  the 
reverse  is  the  case. 

As  it  is  the   quantity  ab  in   Fig.   14   that  we  must 


NOTES   ON   THERMODYNAMICS.  33 

insert  in  the  equation  for  (dp),  we  can  determine  its 
value   from   the   equation  of  the  substance   by  deter- 

lAp  \* 
mining  ljj~J*  which  gives  us  the  rate  of  charge  of  / 

with  T,  and  multiplying  this  by  dt,  or  ab  =  f — —\  dt. 

Similarly  (dv)  —  cd  =  \-^\  dt,  or  writing  these  values 

\  zit  ]p 

in  the  original  equations,  we  have 


//        r 
T  dt :"  'J9 

or  differentiating, 


We  see,  then,  that,  if  heat  is  added  to  any  substance 
along  an  isothermal,  the  quantity  of  this  heat  can  be 
represented  by  either  of  the  two  quantities  in  equa- 
tion (19). 


*This  form  is  chosen  to  clearly  indicate  that  we  wish  to  obtain 
a  number  (or  an  expression)  giving  the  ratio  of  the  simultaneous 
changes  of/  and  7' at  constant  volume,  and  this  in  no  way  de- 
pends on  the  value  of  dt. 


34  NOTES   ON    THERMODYNAMICS. 

Prob.  31.—  Prove  from  equation  (19)  that  if  heat  is  added  to  air 
at   constant    temperature,    the    heat    required    is 


For  air  PV  =  AT  and  [4f\   =  (*£}  .from  this  equa- 

\4t/v     \dt)v 


tion,  gives 

•vdp  =  Rdt, 

dp\        R 


From  equation  (19), 


As  the  temperature  is  to  be  constant,  we  have 


Prob,  32.—  How  much  heat  must  be  added  at  constant  tem 
perature  to  a  substance  whose  equation  is 


6  i  -  273° 
=   I06'1       ~T 


to    change    its  volume   at   constant   temperature 
from  z/i  to  Vi  ? 
Prob.  33.—  Having  given 


,  =        __       _ 
P~~       v    '     TV* 

as  the  relation  between  the  pressure,  volume,  and 
temperature  of  a  substance,  how  much  heat  must 
be  added  at  constant  temperature  to  change  its 
volume  from  Vi  to  7/9,  having  given  the  values  of 
A,  B,pi,  and  T?  . 


NOTES  OAT   THERMODYNAMICS.  35 

If,  however,  the  heat,  instead  of  being  added  along 
an  isothermal,  is  added  along  any  other  line,  the  follow- 
ing method  will  determine  the 
quantity  of  heat.  Let  AB  (Fig. 


1  6)  be  the  line  of  the  expansion, 
the  co-ordinates  being  p  and  v. 
Let  A  and  C  be  points  dt  degrees 
apart.  The  heat  added  between 
the  points  A  and  C  is  represented  FIG.  16.  N 

by  the  area  A^C,  and  this  area  =  dH.  Through  C 
draw  the  isothermal  CD  until  it  cuts  the  line  of  con- 
stant pressure  through  A.  The  heat  added  from  A  to 
C  is  equal  to  that  added  from  A  to  D,  minus  that 
from  D  to  C.  Or,  it  is  more  nearly  true  to  say  that 
the  latter  quantity  becomes  more  and  more  nearly 
equal  to  the  heat  added  from  A  to  Ct  as  the  tempera- 
ture difference  between  A  and  C  becomes  smaller. 
Calling  the  difference  in  temperature  dt,  then  AD  is 


+7-1  dt,  and  CE  is  dp,  as  the  point  C  is  fixed  by  the 

atjt 

intersection  of  the  isothermal  dt  degrees  above  A  and 
the  given  curve  of  expansion  AB.  The  area  $ADi  is 
the  heat  added  from  A  to  D,  or  is  by  definition  cpdt. 
The  heat  from  D  to  C  is  the  area  21  DC,  or,  from 

(/Ji'\ 
-r-  \  dp,  and  we  have  taken  this  form 

because  AD  or  dv  is  the  quantity  fixed  by  the  two 
isothermals  dt  degrees  apart.  The  heat  from  A  to  C  is 


=cpdt-T(-~\  dp  =  dH,  (20) 

' 


3<>  NOTES  OAT  THERMODYNAMICS. 

which    is    one    form    of    the  general    thermodynamic 
equation. 

Another  form  of  this  equation 
is  obtained  as  follows:  Draw  AF 
(Fig.  17)  at  constant  volume  until 
it  cuts  the  isothermal  through  C. 
2  Then  the  area  iAC2  differs  from 
iAF4  +  4FC2  by  the  area  AFC, 
which  disappears  as  dt  is  made 

smaller.     Then 

Ag=dv,          AF=$\dt, 

\AIJV 

and  we  have  the  areas 

AF4i  =  cjt,  4FC2  =  T  (^]dv, 
and 

)dv,.     .     .     .     (21) 

which  is  a  second  form  of  the  fundamental  equation. 
In  these  two  equations  the  terms  — r-  and  — j—  depend 
only  on  the  equation  of  the  substance,  and  could 

have  been  written  -3—  and  —7-,  while  the  other  terms 
at  at 

depend  on  the  law  of  the  expansion.  That  is,  in  the  first 
one  we  have  made  dt  and  dp  depend  on  the  law  which 
we  have  chosen  to  assume  for  the  expansion,  but  the 

Av 
value  -—  depends  only  on  the  substance  which  is  to 


NOTES  ON   THERMODYNAMICS.  37 

expand.  In  using  these  formulae  it  must  be  remem- 
bered that  the  units  must  be  the  same  for  all  the 
terms.  That  is,  if  the  area  dpdv  is  in  foot-pounds,  it 
represents  a  certain  part  of  the  diagram,  and  the  cp  or 
cv  must  be  in  foot-pounds  also  ;  or  if  cp  and  cv  are  in 
heat-units,  the  value  of  dpdv  must  be  in  heat-units 
also. 

Prob.  34.  —  Suppose  that  there  is  a  substance  which,  in  the 
state  we  propose  using  it,  is  a  gas,  and  that  the 
relation  between  its  pressure,  temperature,  and 
volume,  as  determined  by  experiment,  can  be 

T) 

expressed  by  the  equation  pv  =  A  T  --  =.    What 

will  it  do  under  various  methods  of  expanding  it  ? 
First  calling  p  constant,  we  have 


=  Adt 
or 


tdv\  _A       B 

\dt)p-p  +  rp> 

and  calling  v  constant, 


-4_J    B 

r  v  +  rv 


and  the  two  forms  of  the  fundamental   equation 
are  therefore 


(A) 
(B) 


If  now  the  substance  is  to  expand  at  constant  vol- 
ume, we  have,  from  (B),  H  =  cvdt.     If  at  constant 


33  NOTES  ON  THERMODYNAMICS. 

pressure,  from  (A),  //  =  cpdt.     If  at  constant  tem 
perature,  from  (A), 


AT+  ,orf  from  (B), 


If  it  is  to  expand  adiabatically,  we  have  dH  '  —  o  for 

__dp 

both  equations  and  —  =  -  V  »  which  is  in    the 
cv  dv 

/  ^ 

same  form  as  the  equation  for  the  adiabatic  ex- 

pansion of  air. 

In  using  the  fundamental  formula  we  must  remem- 
ber that  the  formula  gives  us  the  heat  added  from  A 
to  B  in  the  figures,  and  that  when  we  speak  of  the 
heat  in  a  substance  we  are  measuring  for  some  datum. 
Ordinarily  this  is  taken  at  32  degrees  F.,  and,  as  this 
is  the  temperature  at  which  a  change  of  state  in  water 
takes  place,  we  must  define  more  particularly,  so  that 
if  we  are  dealing  with  water  or  its  vapor  it  is  cus- 
tomary to  measure  the  heat  from  that  in  water  at  32 
degrees. 

Heat  in  Water  and  Steam.  —  The  application  of  the 
general  formula  to  the  heat  in  a  liquid  and  its  vapor  is 
as  follows  :  When  heat  is  added  to  a  liquid  (water,  for 
instance)  at  32  degrees,  its  temperature  rises  and  its 
volume  changes  slightly.  This  continues  until  the 
temperature  reaches  such  a  point  that  vapor  begins  to 


NOTES   ON   THERMODYNAMICS.  39 

form.  This  is  always  a  definite  point  for  a  given 
pressure.  For  water,  15  pounds  pressure  and  213  de- 
grees correspond,  100  pounds  pressure  and  327  degrees  ; 
for  ammonia,  37.8  pounds  pressure  and  10  degrees, 
1  80  pounds  pressure  and  90  degrees,  etc.  The  addi- 
tion of  any  further  quantity  of  heat  to  the  liquid  which 
is  ready  to  boil  does  not  increase  the  temperature,  but 
vapor  begins  to  form,  part  of  the  heat  being  used  up 
in  increasing  the  volume,  and  part  in  some  sort  of 
internal  work  required  to  change  the  liquid  water  into 
vapor.  This  condition  of  affairs  continues  until  suffi- 
cient heat  has  been  added  to  convert  all  the  liquid 
into  vapor.  Any  further  addition  of  heat  again  raises 
its  temperature  and  continues  to  increase  the  volume. 
The  addition  of  heat,  therefore,  at  constant  pres- 
sure takes  place  in  three  successive  stages  :  first, 
while  it  is  entirely  a  liquid  ;  second,  while  part  is 
liquid  and  part  vapor;  and  third,  after  it  is  entirely  a 
vapor.  We  have  generally 


While  it  is  a  liquid  v  is  practically  constant  and 
dH  =  cvdt,  H  —  cv(Tl  —  T32)  and  is  called  q,  or  the 
heat  of  the  liquid. 

In  reality  there  is  a  certain  amount  of  work  done 
and  dv  is  not  strictly  zero,  but  the  ordinary  value  of 
the  specific  heat  of  liquids  includes  the  very  small 
amount  of  heat  necessary  to  do  the  external  work. 

cv  is  not  necessarily  constant  and  /  cvdt  is  not  neces- 
sarily equal  to  cr(Tj  —  T^).  If  we  know  the  relation 


40  NOTES    ON    THERMODYNAMICS. 

between  cv  and  T,  it  should  be  inserted  before  inte- 
grating and  the  exact  value  found.  It  is  customary 
to  say  that  for  water  cv  =  I,  while  in  reality  c  — 
I  -j-  .00004^  -j-  .0000009/2,  t  being  in  the  centigrade 
scale,  and  we  have 

H  '  —  q  —    i  cdt  —    /  (i  +  .00004/  -f-  .oooooo9/2X/ 

t.y  o  »  /  o 

=  /  +  •  00002  /2  -f-  .  0000003  /3, 

which  is  the  true  value  of  the  heat  of  the  liquid  in 
French  units.  To  get  the  corresponding  quantity  in 
English  units,  enter  this  equation  with  the  centigrade 
temperature,  and  •$•  the  value  of  the  quantity  obtained 
is  the  value  in  B.T.U.  for  the  corresponding  Fahren- 
heit temperature. 

Prob.  35.  —  The  specific  heat  of  liquid  anhydrous  ammonia  is 
given  by  the  equation  (French  units) 


c  = 


How  much  heat  must  be  added  to  i  kilogram  to 

raise  its  temperature  from  20  to  40  degrees  C.  ? 
Prob.  36,  —  What  is  the  specific  heat  of  liquid  ether  at  30  de- 
grees C.  if  the  equation  for  q  (French  units)  is 


q  = 

Prob.  37.  —  How  much  heat  is  required  to  raise  i  pound  of 
water  from  60  to  160  degrees  F.,  using  the  specific 
heat  of  water  ? 

Prob.  38.  —  What  will  be  the  temperature  of  i  pound  of  water 
at  60  degrees  if  10  heat-units  are  added  to  it  ? 

Prob.  39.  —  Using  the  data  of  problem  (34),  how  much  heat  must 
be  added  to  i  pound  of  liquid  ether  to  raise  its 
temperature  from  40  to  50  degrees  F.? 


NOTES   ON   THERMODYNAMICS.  41 

It  is  interesting  to  note  just  what  proportion  of 
this  value  of  q  is  actually  used  for  heating  and  what 
proportion  goes  to  do  outside  work,  because  the  part 
that  does  work  may  or  may  not  be  available  if,  for 
any  reason,  we  have  to  make  use  of  the  heat  in  the 
water. 

One  pound  of  water  at  50  degrees  occupies  .016 
cubic  foot. 

One  po.und  of  water  at  140  degrees  occupies  .01627 
cubic  foot. 

The  amount  of  work  done  if  the  water  is  under, 
say,  100  pounds  pressure  per  square  inch  is  .00027 
X  100  X  144  —  3.89  foot-pounds,  or  .005  heat-units. 
The  total  heat  required  to  raise  I  pound  of  water 
from  50  degrees  F.  to  140  degrees  F.  is  90.1  heat- 
units,  or  a  practically  negligible  amount  is  used  for 
doing  work  and  we  can  say  that  all  the  heat  added 
while  it  is  still  a  liquid  remains  in  it. 

When  the  water  reaches  the  boiling-point  the  tem- 
perature no  longer  rises,  and  we  must  again  apply  our 
general  formula,  as  the  conditions  under  which  it  was 
originally  applied  no  longer  hold.  We  have 


Now 

dt-o     and    H= 

the  total  latent  heat,  as  it  is  called.       As  T  is  con- 
stant, we  could  have  written 


42  NOTES   ON    THERMODYNAMICS. 

To  apply  this  formula  it  is  necessary  to  know  the 
relation  between  p  and  /  for  the  vapor  to  determine 


the  value  of     -jr)    ,  and  it  is  also  necessary  to  know 

the  limiting  values  of  v.  Experimentally,  the  rela- 
tion between/  and  t  can  be  easily  obtained.  The 
value  of  v  when  the  liquid  is  all  vapor  is  difficult  to 
determine  experimentally,  and  as  rcan  be  determined 
readily  by  experiment,  this  formula  is  of  more  value 
in  determining  the  limiting  value  of  v  than  in  deter- 
mining the  value  of  r.  In  applying  the  formula 

dp 

either  way,  we  know  that  -y-  does  not  depend  on  dv, 

as  for  each  pressure  there  is  a  definite  temperature 
and  the  equation  might  have  been  written 


-T(%f*-T  (I)  <».-.'• 

where  vz  is  the  volume  of  I  pound  of  vapor,  and  vl  the 
volume  of  I  pound  of  liquid. 


Prob.  40.  —  What  is  the  volume  of  I  pound  of  saturated  steam 
at  100  pounds  pressure  per  square  inch  if 
r  =  1113.9  —  .695/,  and 

p99       =        99X144,          T99       =326.86+460.7, 

/ioo  =   100X144,       Tioo  =  327.58  +  460.7, 

plol  =  ioi  x  144,     Tin  —  328.30  +  460.7, 

Ap       =     /101—  p99      =    2X144, 

AT  =  7\ui—  T»a  =  1.44, 

rioa  _=  1113.9—  .695  x  327.  58  =  884. 


NOTES  ON   THERMODYNAMICS.  43 

Aft 
From  the  formula  ?•=  T~^(vi—ru^)  we  have 

884  x  778=788.28  x  2  X  I44(z/8— vi), 
1.44 

884  x  778  x  1.44 
or         *'-*"=  2  x. Jx  788.38  =  4'36 

and  z/a  =  4. 36 +  .016  =  4.38. 

Prob.  41. — What  is  the  volume  of  i  kilogram  of  saturated  vapor 

of  ether  at  50°  C.,  using  the  first  five  columns  of 

Table  IV,  Peabody?* 
Prob.  42. — What  is  the  value  of  r  in  English  units  for  carbon 

bisulphide  at  50°  F.,  using  only  columns  i,  2,  3,  9, 

10,  ii  of  Table  VII,  Peabody? 

B         C 

Rankine  gives  log  p  =  A  —  —  —  -=  for  the  rela- 
tion between  the  pressure  and  the  temperature,  and 
the  above  equation  can  be  written 


r  =  p(v<>  — 


Regnault's  experiments  give  the  following  for  the 
relation  between  the  latent  heat  and  the  temperature: 

r  =  1113-9  ~  .695/1 

and  Peabody  has  deduced  constants  for  Regnault's 
formula  in  the  form  of  log/  =  a  —  ban  -4-  cfin  for  the 
relation  between  pressure  and  temperature  which  can 
be  used  for  determining  the  value  of  z/2  —  v^ 

The  value   of  r  above  given  consists   of  two  parts, 
one  of  which  does  external  work  and  the  other  internal 

*  Peabody's  Tables  of  the   Properties  of  Saturated  Steam  and 
other'*  Vapors, 


44  NOTES   ON   THERMODYNAMICS. 

work.  Calling  u  the  difference  in  volume  1*%  —  vi ,  p  the 
pressure,  and  A  the  heat  equivalent  of  work,  the  ex- 
ternal work  is  Apu,  and  the  internal  is  r  —  Apu  =  p. 
The  relations  between  p  and  Apu  are  very  different 
from  the  corresponding  quantities  while  in  a  liquid 
state,  as  the  Apu  is  about  -^p. 

It  is  to  be  remembered  that  the  external  work  has 
been  done,  and  while  the  heat  to~do  it  has  been  ex- 
pended, this  heat  no  longer  exists  in  the  steam  formed. 
It  may  have  been  expended  in  pumping  water,  and 
may  exist  as  potential  energy  stored  in  water  in  some 
distant  reservoir.  That  is,  r  has  been  expended  and  p 
remains  in  the  vapor,  and  the  Apu  is  not  in  the  steam 
and  is  not  available  for  any  future  work.  When  the 
pound  of  water  at  32°  F.  is  heated  and  entirely  evapo- 
rated under  constant  pressure,  we  have  added  to  it 
q  -{-  r  =  A.  heat-units,  and  this  is  called  the  total  heat. 
It  is  often  written  as  total  heat  "  in  the  steam."  This 
expression  is  incorrect,  as  it  is  the  total  heat  required 
to  form  the  steam.  The  amount  of  heat  "in  the 
steam"  is  only  q  -f-  p. 

The  steam  being  entirely  formed,  the  addition  of 
more  heat  at  constant  pressure  superheats  it,  and  it 
has  been  found  that  the  specific  heat  of  superheated 
steam  at  constant  pressure  is  .48.  That  is,  if  the 
steam  is  raised  /  degrees  above  its  point  of  saturation 
the  heat  added  is  .48*  —  ^(^Sup.  —  ^sat.)- 

Of  this  heat  added,  only  a  portion  remains  in  the 
steam.  A  certain  amount  of  external  work  must  be 
done,  and  while  we  have  expended  .48(7"sup  —  ^sat.) 
heat-units,  a  quantity  of  work  has  been  done  equal  to 


NOTES   ON   THERMODYNAMICS.  45 

/(^sup.  —  z'sat.)-      The  heat  remaining  in  the   steam  is 
therefore 


To  determine  the  value  of  this  quantity  we  must 
have  the  relation  between  the  pressure,  volume,  and 
temperature  of  superheated  steam. 

This  relation  determined  experimentally  can  be  ex- 
pressed by  the  following  equation  (Peabody)  : 

/^  =  93.5r-  97  1/*, 

from  which  either  T  or  v  can  be  readily  found  if  the 
remaining  two  quantities  are  given.  In  tabular  form 
we  then  have,  starting  with  water  at  32  degrees  and 
ending  at  the  state  given  below  : 

ALL    LIQUID. 

Heat  added  —  q\ 
Heat  remaining  =  q. 

MIXTURE    OF    LIQUID    AND    VAPOR. 
(x  =  parts  vapor.) 

Heat  added  =  q  +  xr\ 
Heat  remaining  =  q  +  xp  ; 
Work  done  —  xApu. 

ALL   VAPOR. 

Heat  added  =  q  +  r  +  .4*(Tmp.  -  Tsat)  ; 
Heat  remainin 


Work  done—  Apu-\-p(v^  ~  ^at. 


46  NOTES   ON    THERMODYNAMICS. 

Prob.  43. — How  much  external  work   is   done  in  converting  I 
pound  of  water  at  60  degrees  into  a  mixture  hav- 
ing x  =  .6  at  150  pounds  pressure? 
How  much  heat  is  expended  ? 

Prob.  44. — How  much  heat  is  in  I  pound  of  superheated  steam 
at  150  pounds  pressure  and  400  degrees  F.,  count- 
ing from  32  degrees,  and  how  much  work  has  been 
done  ? 

Prob.  45. — If  80,000  foot-pouncls  of  external  work  is  done  in 
converting  i  pound  of  water  into  steam  at  150 
pounds  pressure,  what  must  be  the  condition  of 
the  steam  ? 

The   distinction  between   the  heat   added   and   the 
heat  remaining  in  a  substance  can  be  perhaps  better 
understood   by   the   following   example :     Suppose   B 
(Fig.  1 8)  is  the   initial   state   of   I 
pound  of  water  at  15  pounds  pres- 
A         sure  and  213  degrees  F.,  and  A  is 

Iits  final  condition   at    100  pounds 
pressure   and   327.58    degrees    F. 
c          At  B  the  water  has  in  it  q—  1 8 1. 8 


; heat-units.      At  A    it  will  have  in 

FIG.  18. 

it  as  steam 

qJfp=i  297.9  +  802.8  =  1 100.7. 

To  pass  from  B  to  A  we  must  do  a  certain  amount  of 
work.      The  difference  in  volume  between  B  and  A  \ 
4.387  cubic  feet. 

Suppose  that  the  volume  A  is  first  filled  at  1 5  pounds 
pressure,  and  that  afterwards  heat   is  added  and  the 


NOTES   ON    THERMODYNAMICS.  tf 

pressure  is  raised  to    100  pounds  from   C  to  A  :  the 
amount  of  work  done  is  equal  to 

15  x  144  x  4-387 

-  —  =  12.2  heat-units. 

The  total  heat  that  must  be  expended  is  therefore 
1 100.7  +  I2-2  —  1 8 1. 8  =  931.1  heat-units. 

Suppose  again  that  the  pressure  is  first  raised  to  D 
and  the  volume  is  then  increased  to  A.  The  work 
done  in  this  case  is  equal  to 

IPO  X  144  X  4-387  , 

— 3 —    =  81.2  heat-units, 

775 

and  the  heat  required  is 

1 100.7  +  81.2  —  1 8 1. 8  —  1000.1. 

It  is  therefore  to  be  noted  that  the  amount  of  heat 
which  must  be  expended  depends  upon  the  way  in 
which  it  is  expended,  but  that  the  portion  of  the  heat 
added  which  remains  in  the  substance  is,  in  the  exam- 
ple above  given,  always 
• 

1100.7  —  181.8  =  918.9  heat-units, 
or 

0100  +   ^100   -    ?15' 


48  NOTES   ON    THERMODYNAMICS. 

Prob,  46. — Four  pounds  of  a  mixture  of  steam  and  water  at  60 
pounds  pressure  per  square  inch  fill  a  vessel  A  of 
10  cubic  feet  capacity,  and  6  pounds  of  mixture  fill 
another  vessel,  B,  of  10  cubic  feet  at  100  pounds 
pressure.  If  the  contents  of  the  two  vessels  are 
intimately  mixed,  the  volume  not  changing,  wrnt 
will  be  the  final  pressure,  assuming  no  radiation  ? 
First  determine  the  heat  in  vessel  A.  We  have 

4_*r  x  7.096  +  4(1  —  x)  .016  =  10,     x  —  .35. 

-  Heat  =  4(261.9  +  .35  x  830.7)  =2212. 
To  determine  the  heat  in  vessel  B  : 

6x  x  4.403  +  6(1  —  x)  .016  =10,     x  =  .376. 
Heat  =  6(297.9  +  .376  x  802.8)  =  3600, 
The  heat  per  pound  of  the  mixture  is  then 

22,2  +  3600  = 
10 

and  the  volume  occupied  per  pound  is  f &  =  2  cubic 
feet.     We  have  then  two  equations  to  satisfy : 

x  x  s  +  (i  —  ;r).oi6  =  2, 
q  +  *P  =  58l-2> 

and  these  can  best  be  solved  by  trial. 

Prob,  47. — What  heat  must  be  added  at  constant  volume  to 
raise  the  pressure  of  one  pound  of  a  mixture  of 
steam  and  water  occupying  3.8  cubic  feet  from 
100  to  150  pounds  pressure  per  square  inch  ? 

Prob.  48. — A  vessel  of  10  cubic  feet  capacity  has  in  it  4  pounds 
of  a  mixture  of  steam  and  water  at  100  pounds 
pressure ;  25  pounds  of  water  at  60  degrees  F.  are 
pumped  into  the  vessel.  What  is  the  resulting 
temperature,  assuming  no  radiation? 

Prob.  49. — If  10  cubic  feet  of  dry  saturated  steam  at  100  pounds 
pressure  per  square  inch  is  allowed  to  pass  from  a 


NOTES  ON    THEKMODYNAMrCS. 


49 


boiler  into  an  open  vessel  having  in  it  25  pounds 
of  water  at  60  degrees  F.,  what  is  the  resulting 
temperature  ? 

Adiabatics. — We  have  already  proved  that  if  a  sub- 
stance expands  at  constant  tenv" 
perature  between  two  adiabatics, 
the  heat  added  divided  by  the  tern,-  jjj 
perature  is  constant.     To  repeat  in  2 
as  lightly  different  form,  let  the  dia-  % 
gram,  Fig.  19,  be  a  heat  diagram, 
in  which  AB  and  EF  are  constant- 
temperature  lines,  and  A  C  and  BD 
are  two  adiabatics.     Then  the  area  ABDC  divided  by 

•         TT  TT 

rr*  /->  7-k  /~*      i  •     •  i       t     i           T>                     J^  AB  —  •"  EF   J  • 
A  =  area  EFDC  divided  by  T E ,   or  -=- ^r-  dl~ 


ENTROPY 

FlG*  I9' 


rectly  from  the  figure.     We  can  also  write 

/A  f*A  f*E 

dff_     I    djf_     I    dH_ 
~T~JF  ~T~JB  ~r" 

as  each  of  these  quantities  is  the  horizontal  distance 
between  the  lines  AC  and  BD.  That  is,  it  makes  no 
difference  how  much  heat  is  added  between  E  and  B, 

for  instance,  nor  how  it  is  added,  the  quantity     /  —  =- 

H  PT 

is  constant  and,  if  we  please,  is  equal  to 


or  is  equal  to 


fBj(ffEGB) 

Jz  '      T 


or 


the    T  in  the  latter  case 


being    a  variable,   and    is  the    temperature    at  which 
is  added. 


5O  NOTES   ON    THERMODYNAMICS. 

Along  theadiabatic,  as  dH  =  o,  we  have     /  — .=0. 

/dH 
-^-  is  constant  between  two 

adiabatics   for  any  substance  gives  us 
another  method  of  obtaining  the  equa- 

<F~ c    tion  to  the  adiabatic  for  air.      In  Fig. 

20   suppose   a   to   be   a   point   on   one 


FIG  20.          adiabatic,  and  b  and  ^points  on  another. 

CdH  . 
As     /   —  is  constant  from  a  to  b,  or  to  <:,  suppose  ab 

to  be  a  constant-volume  line  and  ac  a  constant-pres- 
sure line. 

We  have  for  ab 


CdH 
J  ~Y"' 


for  ac 

rdH=  fr'cj#=    lop  T, 

and 

Tt  Tc 


NOTES   ON    THERMODYNAMICS. 


or,  as/z'  =  RT, 


\v>     A1-41^1-41' 


or 


which  we  have  before  deduced  in  an  entirely  different 
way. 

When  we  come  to  apply  this  method  to  liquids  and 
vapors  the  problem  is  rather  more  complicated.  In 
Fig.  21  suppose  a  to  represent  the 
pressure  and  volume  of  I  pound  of 
water,  and  suppose  the  temperature 
to  be  7\.  Let  be  be  an  adiabatic 
curve  such  that  at  b  we  have  xb 
pounds  of  steam  and  I  —  ^pounds 
of  water,  and  suppose  that  at  c  we 
have  xe  pounds  of  steam  and  I  —  xc  pounds  of  water. 
We  know  that 


FIG.  21. 


r*H    r 

/  ~r" ''  / 

tS  a        *  t/  a 


dH 


from  what  has  just  been  proved. 

On  the  path  from  a  to  b  suppose  first  the  tempera- 
ture is  raised  to  d,  and  then  that  xb  pounds  of  steam 
are  made.  From  a  to  d,  dH  =  cdt,  because  in  the 
general  formula, 

dH  =  c.4t  4- 


we  have  dv  =  o,  and  hence     / 


H 
T 


52  NOTES   ON    THERMODYNAMICS. 

From  d  to  b  the  heat  dH  added  is  rdx,  and  we  can 
write 


d 
and 


/r  rb 
off  _     I     cat 
T    "  '  IT     T 
«  */      a        •*• 

From  a  to  r  we  can  write 


^- 

T     'Jr. 


4_ 

'    ' 


T.    T        Tf' 
and  as  these  are  equal,  we  can  write 

/  V /+/J /  ->*     >*  f  .  '     *  /*/T  /  -V    <>* 

/         £ff    ,    -3y*_     /        fff    •    ££«  /,,-N 

l'r~T~T~~7F''^~T~l~~Tr'       '       (23) 

*yy  •'b       «yya-*          * « 

In  this  equation  ^r  is  the  specific  heat  of  water,  or  is 

^ 

-77,   and   if  we   know   one   value  of  x,  we   can   deter- 

at 

mine  any  other.      Ordinarily  the  value  of     /  —  can 

be  calculated  with  sufficient  accuracy  by  calling  <:  =  I, 

////  T 

—  is  then  c  loge—?;  but  Peabody's  tables  give 

the  value  of  this  quantity  using  the  exact  value  of  c, 
so  that  it  need  not  be  calculated. 


NOTES   ON    THERMODYNAMICS.  53 

Prob,  50,  — If  i  pound  of  a  mixture  of  steam  and  water  occupying 
3.8  cubic  feet  at  a  pressure  of  100  pounds  absolute 
expands  adiabatically  to  15  pounds  pressure,  what 
is  its  volume  ? 

We  have  from  the  steam-table 
7\0o  =  327.58  +  460.7;  Vol.  i  Ib.  steamioo  =  4403  cu.  ft.; 

—  Q  Q  A 

Calling  7«  =  32°  +  460.7,  we  have 
-y-  =  2.3026  (loj  788.28  —  log  492.7)  =  470 

approximately,  or  .4733  from  the  tables. 
r1B  =  213.03  4-  460.7;  Vol.  i  Ib.  steam  is  =  26.15  cu-  ft-J 
fcdl 

Klf    =     965.^      J       —=.3143. 

To  determine  x\>  we  have,  as  .016  is  the  volume  of 
a  pound  of  water, 

(i  —  xb}  .016 +xb  4403  =  3'8  ; 

We  can  then  write 

'T™cdt_      .861  x  884  _    PT"cdt     ^965.1 

r~"     788.28    ~~JTn  '^r+  673^3  ; 

4733  +  .964=  .3143  +  1431^; 

xc  =  .782  ; 

Vol.  =  .782  x  26.15  4-  (i  —  .782)  .016  =20.5  cu.  ft. 
Prob,  51 1 — If  i  pound  of  a  mixture  containing  40  per  cent  of 
water  is  compressed    adiabatically  from  20  to  60 
pounds  pressure,  what  is  the  percentage  of  mois- 
ture at  the  higher  pressure? 

Prob.  52. — A  pound  of  a  mixture  is  expanded  adiabatically,  so 
that  it  has  the  same  percentage  of  water  at  60  and 
15  pounds.  What  must  have  been  the  percentage 
at  60  pounds  pressure? 


54  NOTES   ON   THERMODYNAMICS. 

Whenever  a  body  expands  adiabatically,  or  at  the 
expense  of  its  own  heat,  the  amount  of  external  work 
done  must  be  the  difference  in  the  quantity  of  heat  in 
it  at  the  beginning  and  at  the  end  of  the  expansion. 

If  we  have  a  mixture  of  steam  and  water  at  the 
beginning  of  the  expansion  so  that  the  portion  of 
steam  is  xlt  the  heat  present  is  ql  -\-  x^pl.  At  the  end 
of  the  expansion  the  heat  is  q^  +  ;r2p2,  and  the  amount 
of  work  done  is  therefore 


Prob.   53. — In    problem  (50)  how  much  work  is   done  in   the 

expansion  ? 
From  the  tables 

?i  =  297.9,      pi  =802.8, 
^2  =  181.8,      pi  =  892. 6,      and 

the  work  =  297.9  +.861  x  802.8  —  181.8  —.782  x  892.6  =  107  h.  u., 
or  107  x  778  =  83200  ft.-lbs. 

Prob.  54. — What  work  is  done  if  20  cubic  feet  of  a  water  mix- 
ture weighing  6  pounds  expands  adiabatically  from 
80  pounds  to  20  pounds  pressure  ? 

Prob.  55. — i  pound  of  steam  at  100  pounds  pressure  expands 
adiabatically  to  15  pounds.  How  much  work  is 
done? 

Prob.  56. — i  pound  of  water  at  327  degrees  F.  expands  adia- 
batically to  15  pounds  pressure.  How  much  work 
is  done  ? 

If  we  are  dealing  with  superheated  steam  instead  of 
3.  mixture,  we  have  for  the  value  of  /  —=-  three  parts: 


NOTES   ON    THERMODYNAMICS.  55 

cdt 


one 


ct 
while  it  is  still  a  liquid  or  /    —  ^-,  one   while  it  is 

becoming  steam  at  constant  temperature,  or  —  (as  it 
is  all  converted  into  steam),  and  a  third  portion, 


f, 


7sat.         2 


and  we  can  write 

CdH          Ccdt 


When  superheated  steam  expands  adiabatically,  we 
have,  for  the  amount  of  work  done,  the  difference  in 
the  quantity  of  heat  at  the  beginning  and  end  of 
expansion. 

The  heat  at  the  beginning  is 


, 

4l  +P  1 

The  heat  at  the  end  of  expansion  is 

,r  cr     T  \  A(^.UP.  --  F-QI 

P2    +       ^(^sup.    --    7*t.)    ~  , 


on  the  assumption  that  it  remains  superheated  until 
the  end  of  the  expansion. 

Prob.  57, — i  pound  of  steam  at  150  pounds  pressure  occupies  a 
volume  of  3.3  cubic  feet.  What  is  its  condition 
after  it  expands  adiabatically  to  15  pounds  pres- 
sure,  and  what  work  is  done'?  ^^^R  A  R>^ 

UN1VEK      rx/ 


56  NOTES   ON    THERMODYNAMICS. 

As  i  pound  of  saturated  steam  at  150  pounds  pressure 
occupies  3.011  cubic  feet,  the  steam  in  the  problem 
must  be  superheated,  and  from  the  equation  of 
superheated  steam  we  have 


93-5      ' 

or 

=  I5ox  144x3.3  +  97i  xdsox  144)*    go 
93-5 

Saturated  steam  at  150  pounds  pressure  has  a  tempera- 
ture of  358.26  degrees  F.  =  818.96,  or  the  steam  is 
superheated  71  degrees.  We  have  then 

dH  cdt       r  80 


861.2  890 

-        +  -48  x  2.3026  log 


=  1.6055. 
At  15  pounds  pressure 

>cdt  r 


As  the  sum  of  these  two  =  1.7473  is  greater  than 
1.6055,  tne  steam  is  evidently  not  superheated  at 
the  lower  pressure  and  we  have 

1.6055  =  -S^  +  ^XMSS. 
x  =  .901. 

To  determine  the  amount  of  work  done  during  this 
expansion,  we  have  the  heat  at  the  initial  condition 

=   ?I60+/°,50  +    I     ^(890-8l9)-         ^g44    (3.3-3.0II)      I 

[Heat  added  less  work  done] 
=  330  +  778.1+26.07  =  1134.17 


NOTES   ON    THERMODYNAMICS.  S7 

At  the  final  condition  the  heat  in  the  steam  is 
?18  +  .90i/oIB  =  i8i.8  +  .90i  x  892.6  =  985.03. 
The  work  done  is 

1134.17  —  985.03  =  149.14  h.  u.  =  116000  ft.-lbs. 

Prob.  58, — If  in  the  above  problem  the  volume  had  been  4  cubic 
feet  at  150  pounds  pressure,  what  would  have  been 
the  condition  and  how  much  work  would  have 
been  done  if  it  had  expanded  to  15  pounds  pres- 
sure ? 

Prob.  59. — If  i  pound  of  steam  at  15  pounds  pressure  super- 
heated 60  degrees  is  adiabatically  compressed  to 
100  pounds  pressure,  what  is  its  temperature  and 
volume? 

Curve  of  Constant  Steam  Weight. — If   I   pound   of 
saturated  steam    expands  in  such  a  manner  that  we 
have  always  I  pound  of  saturated  steam  whatever  its 
pressure,  the  expansion  curve  is  called  a  curve  of  con- 
stant steam  weight.     Or  if  a  mixture  of  steam  and 
water  having  a  given  proportion  of  steam  expands  in 
such  a  way  that,  whatever  its  pressure,  there  is  always 
the  same  proportion  of  steam  present,  the  curve  of 
expansion  is  called  a  curve  of  constant  steam  weight. 
Prob.  60, — If  i  pound  of  a  mixture  of  steam  and  water  at  120 
pounds  pressure  expands  so  that  30  per  cent  is 
always  steam,  what  are  the  volumes  at  120,  90,  60, 
and  30  pounds  pressure  ? 

At  120  pounds  we  have  for  the  volume  of  the  steam 
.30x3.711,  and  for  the  water  .7ox.oi6,  and  the  total 
volume  is  1.1133  +  . 0112  =  1-1245  cubic  feet. 
Prob,  61, — A  mixture  of  60  per  cent  steam  and  40  percent  water 
expands  from  90  to  15  pounds  pressure,  so  that 
there  is  always  60  per  cent  steam.  What  is  the 
volume  at  every  15  pounds  pressure,  if  the  total 
weight  is  5  pounds? 


5§  NOTES   ON   THERMODYNAMICS. 

To  Determine  the  Work  Done.  —  The  amount  of  work 
done  by  such  an  expansion  can  only  be  approximately 
determined  by  calculation.  The  most  convenient  way 
of  doing  it  is  to  assume  that  the  expansion  curve  is  in 
the  form  pvn=K'  ,  and  find  the  most  probable  value  of 
ny  and  from  the  equation  of  the  curve  determine  the 
area. 

To  determine  the  most  probable  value  of  n,  it  is 
not  correct  to  determine  several  values  of  n  and 
average  them.  The  following,  from  the  method  of 
least  squares,  gives  the  most  probable  value  of  n  and 
is  not  at  all  difficult  to  follow  out.  Determine  as 
many  values  of  /  and  v  as  desired,  and  write  these 
values  in  the  logarithmic  equation  as  below  : 


>  —  K"  J 

+  n  log  ^2  =         K"  \ 
+  n  log  7'3  =         K",  etc. 

Add  these  equations  together  and  we  have 

(A) 


Now  multiply  each  of  the  original  equations  by  the 
coefficient  of  n  in  that  equation  and  we  have 

log  A  log  z/j  +  n  (log  i\  )z  =  K"  log  vi  5 

log/2  log  V*  +  n   (10S  ^2  )2  —   K"  log  V*\ 

log/3  log  vz  +  n  (log  vz  )2  —  K"  log  vs. 
Adding  these  equations  together  we  have 

2  log  /  log  v  +  n2  (log  vf  =  ZK"  log  7-.    .    (B) 


NOTES   ON    THERMODYNAMICS.  59 

Solving  (A)  and  (B)  will  give  the  most  probable  value 
of  n.  Ordinarily  three-place  logarithms  are  not  accu- 
rate enough  for  this  work.  The  amount  of  work 
is  then 


I  —  n 

To  determine  the  quantity  of  heat  that  will  be  re- 
quired to  produce  this  expansion,  we  know  that  the 
heat  at  the  end  of  the  expansion  added  to  the  work 
done  must  be  equal  to  the  heat  in  the  steam  at  the 
beginning  of  the  expansion  added  to  the  heat  sup- 
plied. We  have  already  shown  how  to  determine 
three  of  these  quantities  so  that  the  heat  supplied  can 
be  determined. 

Prob.  62. — i  pound  of  steam  at  60  pounds  pressure  expands  to 
40  pounds  along  a  curve  of  constant  steam 
weight.  How  much  work  is  done  and  how  much 
heat  must  be  supplied  ?  We  have  the  following 
for  the  pressures  and  volumes  : 

At  60  Ibs.  V  =  7.096  cu.  ft.; 
50  Ibs.  F=  8.414  cu.  ft.; 
40  Ibs.  V  =  10.37  cu.  ft. 

To  determine  the  law  of  expansion  write  : 

log p  +  n  log  v  =    K" 

1.778+   .851;*    =    K"  1.513  +   .724^=    .851^" 

1*099+   .925;*    =    K"  1.572  +   .856;;=    .925/4"' 

1.602  +  1.016;;    =    K"  1.628  +  1.032;;  =  i.oi6A^" 

5.079  +  2.792;;    =  $K"  (A)  4.713  +  2.6i2«  =  2.792A"'  (B) 

n  =  1.07. 

Work  =  60x144x7.09^-40x144x10.37  =  2  ft   lbs> 

1.07  —  i 


60  NOTES   ON    THERMODYNAMICS. 

21500 
Heat  required  =  q**  +  p*o  +  -^-~o  --  ?60  ~~  PM> 

236.4  +  850.3  +  27.6  —  261.9  —  83°-7  —  2I-7  n-  u- 

Rectangular  Hyperbola.  —  In  many  cases  a  rectan- 
gular hyperbola  practically  represents  the  expansion 
taking  place  in  a  mixture  of  steam  and  water  under 
actual  conditions.  This  is  in  no  sense  a  theoretical 
expansion  line  for  a  steam  expansion,  but  it  practi- 
cally represents  what  actually  takes  place  in  many 
steam-engine  cylinders.  The  law  of  the  expansion 
here  is/z/  =  Ky  and  the  amount  of  work  done  is 


The  amount  of  heat  required  is 


e  Ml  —  q\  —  *iPi, 
\ 


the  subscript  2  referring  to  the  final  condition,  and  I 
to  the  initial  condition. 

Prob.  63.  —  i  pound  of  a  water  mixture  containing  30  per  cent 
of  moisture  expands  from  100  pounds  to  20 
pounds,  so  that  30  per  cent  of  moisture  is  always 
present.  How  much  work  is  done,  and  must 
heat  be  added  or  taken  away,  and  how  much  ? 

Prob.  64.  —  i  pound  of  a  mixture  containing  30  per  cent  of 
moisture  expands  from  100  pounds  to  30  pounds 
along  a  rectangular  hyperbola.  How  much 
work  is  done,  what  is  the  condition  at  the  end  of 
the  expansion,  and  how  much  heat  must  be 
added  or  taken  away  ? 


NOTES   ON    THERMODYNAMICS.  6 1 


CYCLES  PASSED  THROUGH  BY  VAPORS. 

When  a  vapor  is  used  in  a  cylinder  the  amount  of 
work  done  and  the  amount  of  heat  required  can  be 
determined  as  follows:  Suppose  that  at  #,  Fig.  22,  we 
have  I  pound  of  a  mixture  of  vapor  and  liquid,  xa  parts 
being  vapor,  and  suppose  that,  at 
b,  xb  parts  are  vapor,  the  pressure  re- 
maining constant. 

From  b  to  c  let  the  expansion   be 
according  to  any  law,  and  at  c  let  xc 

be  the  proportion  of  the  vapor.     Let  a'  _?'       b'    c' 

FIG.  22. 
heat  be  taken  away  first  at  constant 

pressure,  and  then  according  to  the  same  law  as  the 
expansion  curve  be,  so  that  we  have  at  the  end  of  the 
cycle  the  same  condition  of  affairs  as  at  the  beginning. 
The  amount  of  work  done  is  the  area  of  the  figure 
abed.  It  can  be  most  easily  calculated  by  finding  the 
separate  areas  and  combining  them  so  that 

W  =  abb'  a'  +  bcc'b'  -  cdd'c'  -  add'  a'. 

1^*7**** 

The  area  abb'  a'  =  (xb  —  x^)Apaua. 

~^\    >  -  ^  Z.  ^** 

The  area  cdd'c'  =  (xc  —  x^)Apcuc. 


areas  under  be  and  ad  depend  upon  the  law  of 
the  expansion  and  can  be  determined  as  shown  before. 
The  amount  of  heat  required  to  do  this  work  is  the 
heat  required  from  a  to  c  and  is  equal  to 

(&  +  *cpc)  -   (3  a  +  *a?a)  + 


62  NOTES    ON    THERMODYNAMICS. 

\     The  amount  of  heat  which  must  be  taken  away  is 

—  (qa  +  *apa]  +  adcc'a'  +  (qc  +  xcpc). 
The  relation  between  the  various  values  of  x  depends 
on  the  law  of  the  expansion. 

If  the  expansion  is  adiabatic,  the  value  of  xc  and  xd 
can  be  determined  if  xa  and  xb  are  given.      We  have 


s. 


cdt       xbrb       xcrc 


Tt'       Tc 
all  tl  2  terms  of  which  are  known  except  xc;   and 


xdrd^ 


Jd    T         Ta   '       Td 

from  which  xd  can  be  determined. 

As  Yb  and  Tb  are  equal  to  Ya  and  Ta,  and  similarly 
for  c  and  d,  we  can  write  from  the  last  two  equations 

r  a  YC 

\%6  % a)  y      v*<  •*<£/• 

a  -L  c 

The  work  done  can  then  be  written 

W=  Apbub(xb  —  xa)  +  \qb  +  xbpb  —  qc—  xcp^\ 

-  Apciic(xc  —  x^)  —  \qa  +  xapa  —  qd  —  xdpd~\ 

=  xbYb  —  xaYb  —  xcrc  +  xdYc 
—  (xb  —  xa}rb  —  (xc  —  xd)rc 


—  (xb  -  x*)ra 


NOTES   ON   THERMODYNAMICS.  63 

In  the  last  equation  (x  —  #a)  ra  is  the  heat  added 

Ta—  Tc 
from  a  to  b.     The  efficiency  is  therefore     a  '    —  c,  which 

a 

is  Carnot's  efficiency,  as  might  have  been  expected  as 
this  is  a  Carnot  cycle.  When  this  condition  of  affairs 
exists  in  a  cylinder,  the  cylinder  fulfils  the  functions 
of  boiler,  engine,  and  condenser,  as  we  have  assumed 
that  the  given  weight  of  the  substance  is  in  the 
cylinder  at  all  the  points  of  the  cycle. 

Prob,  65.  —  How  much  work  is  done  in  the  cycle  of  Fig.  3$,  if  5 
pounds  of  a  mixture  of  steam  and  water  expands 
having  pa  =  100  pounds  per  square  inch,  pa  =15 
pounds  per  square  inch,;ra  =  .1,  xb  —  .9  ? 
From  the  tables 

;  a  =  884.0  h.  u.  ;      Ta  —  788.3  ;      Td  =  673.7. 
The  heat  added  from  a  to  b  is 

M(xb  -  xa  }ra  =  5  x  (.9  -  .  i)  x  884.0  =  3536  h.  u. 
The  work  done  is 


=40200°  t'- 


Prob,  66.  —  i  pound  of  NH3  expands  through  a  cycle,  as  in 
Fig.  22.  If  /a  =  60  degrees  F.,  id  =  10  degrees  F., 
xa  —  .1,  x\  —  i,  how  much  work  is  done  and  how 
much  heat  is  required  ? 

Prob.  67.  —  If  in  a  cycle,  like  Fig.  22,  ve  =  10  cubic  feet,  va  =  i 
cubic  foot,/a  =  1  50  pounds  per  square  inch,  pc-—  15, 
how  much  work  is  done  and  how  much  heat  is 
required  if  i  pound  of  steam  is  used? 

In  an  actual  engine  the  conditions  are  different 
from  those  in  the  last  figure,  as  from  a  to  b  there  is 
not  the  same  weight  in  the  cylinder,  and  from  c  to  d 


64  NOTES   ON    THERMODYNAMICS. 

the  weight  also  varies.  And  in  addition  there  is  con- 
stant interchange  of  heat  between  the  cylinder  walls 
and  the  steam. 

First,  neglecting  the  action  of  the  cylinder-walls, 
suppose  Fig.  23  represents  what  takes 
place  in  the  cylinder.  At  a  the 
clearance  volume  is  filled  with  steam 
whose  steam  proportion  is  xa.  The 
steam  from  the  boiler  is  admitted 
and  fills  the  cylinder  to  c.  Expan- 

rlG.    23. 

sion  takes  place  to  d,  and  exhausts 
to  a  again. 

Let  m  pounds  be  in  the  cylinder  at  a,  and  M  pounds 
be  added  from  the  boiler.  Let  x'  be  the  value  for 
the  steam  coming  from  the  boiler.  If  we  know  the 

v 

volume  at  c,  we  have  —    -^>  —  volume  of   I   pound 

m  -f-  M 

and 


from  which  xe  can  be  found. 

To  find  the  work  from  b  to  c  we  have  that  the  heat 
at  c  added  to  the  work  done  is  equal  to  the  heat  at  a 
added  to  the  heat  received  from  the  boiler,  or 


Work  be  =  m(qa  +  xapa  -  qc  -  xcpc)  +  M(x're  -  *>,). 
We  might  have  written 


NOTES   ON   THERMODYNAMICS.  65 

but  it  has  been  written  in  the  form  first  given  to  call 
attention  to  the  fact  that  the  last  term  in  the  first 
equation  contains  rc  and  not  pc.  The  reason  is  that 
the  heat  brought  into  the  cylinder  from  the  boilers 
includes  not  only  qc  and  x'  '  pc,  but  also  the  external 
work  which  must  be  done  in  forcing  this  steam  out  of 
the  boiler,  or  x  'Apcuc. 

The  work  under  cd  is  determined  as  before  shown, 
and  the  work  under  da  is  the  area  of  the  rectangle 
under  ad,  or 


Prob.  68.  —  In  an  engine  having  Fig.  23  for  a  card,  let  Va  =  .4 
cubic   feet,     V*  =  8   cubic   feet,  pb  =   100  x  144, 
pa  =  1  5  x  144,  xa  =  .9,  Xd  =  -8,  cd  being  an  adia- 
batic.     How  much  work  is  done  ? 
First  find  xc. 


/1 


cdt^      JCcTmo  _  .8r16> 
T        T^ioo         T\* 


To  find  the  volume  at  c  we  must  know  the  weight 
along  cd  and  we  have 


i6  =  8. 
From  the  tables  s*  —  26.15  I 

^  =  .8X26.15  +  .  2X.OI6  =  -^2  =  '3Sl  lbS'; 

Vc  =  .381  (.88x4.403  +  .12  x.oi6)  =  1.47  cu.  ft. 


66        NO  TES  ON  THE R MOD  YNA  MICS. 

The  work  ^  =  (1.47  —  4)100x144  =15400 

The  work  cd  =.  . 381^100  +  . 88p100  —  q™  —  .8p15]x778  =  32000 

474°o 

The  work  da  =  (8  —  .4)15  x  144  =  16400 

Work  in  cycle  =  31000 

Prob.  69. — In  the  above  problem,  how  much  steam  must  the 

boiler  have  furnished  if  x'  =  i  ? 

Prob.  70. — How  much  steam  was  in  the  cylinder  at  b,  and  what 
was  the  value  of  Xb  if  there  was  no  loss  of  heat 
through  or  to  the  cylinder-walls  ? 

Prob.  71. — In  problem  68,  how  much  heat  must  have  been 
taken  up  by  the  cylinder-walls  if  x'  =  i  and 
xc  =  .88  ? 


If,  instead  of  the  exhaust  continuing  to  a,  it  had 
stopped  at  e  of  Fig.  24,  the  above  formula  will  apply 
by  putting  in  the  corresponding 
values  of  pressures  and  temperatures, 
etc.,  for  the  new  point  a,  and  the 
amount  of  work  will  be  reduced  by 
the  area  aef,  which  must  be  deter- 


mined as  already  shown. 
In  all  engines  using  vapors,  the  quantity  of  heat  re- 
jected along  the  line  da  of  Fig.  23,  or  de  of  Fig.  24, 
is  a  large  proportion  of  the  total  heat  supplied  to  an 
engine.  To  use  the  same  working  substance  over  and 
over  again  in  an  engine,  it  must  be  liquefied,  pumped 
into  a  boiler,  and  evaporated  again.  All  the  heat  re- 
jected from  the  engine  less  the  amount  which  remains 
in  the  working  substance  as  a  liquid  cannot  be  again 
utilized  for  doing  work  in  the  same  engine.  The 
quantity  of  heat  which  must  be  supplied  to  the  work- 
ing substance  for  each  cycle  is  therefore  the  amount 


NOTES   ON    THERMODYNAMICS.  6? 

which  must  be  added  to  it  as  a  liquid  at  the  tempera- 
ture of  its  discharge  from  the  engine. 

Prob.  72.  —  In  Fig.  24,  using  steam,  if  xc  =  .7,  mc  =  i,  xe  =  it 
me  =.itpb  =  ioo  x  144,  ^=15x144,  /a  =  3ox  144, 
the  curves  cd  and  ea  are  rectangular  hyperbolas, 
how  much  work  is  done  per  cycle  and  how  much 
heat  is  expended  ? 

Prob.  73.  —  If,  having  given  the  data  of  problem  72,  the  sub- 
stance is  anhydrous  ammonia,  what  work  is  done 
and  how  much  heat  is  expended  per  cycle  ? 

Prob.  74.  —  If,  having  given  the  data  of  problem  72,  the  sub- 
stance is  SO2,what  is  the  work  done  and  what  the 
heat  expended  per  cycle  ? 

When  the  action  of  the  cylinder-walls  is  taken  into 
account,  the  following  analysis  might  be  made  after 
the  method  of  Hirn.  Assume  that  at  c,  Fig.  23,  we 
have  steam  with  a  given  proportion  of  moisture  and 
that  the  expansion  is  a  rectangular  hyperbola,  and 
assume  further  that  saturated  steam  without  moisture 
has  been  supplied,  which  is  nearly  true,  and  that  the 
steam  discharged  is  steam  without  moisture,  which 
may  or  may  not  be  true. 

From  c  to  d  the  cylinder-walls  must  give  up  heat  per 
pound  equal  to 


all  the  terms  of  which  are  known   except  xd.      This 
can  be  calculated  from 


68  NOTES   ON    THERMODYNAMICS. 

From  d  to  a  the  cylinder-walls  must  give  up  heat  to 
the  amount 


ra)  -    a 


This  is,  of  course,  on  the  assumption  that  no  heat 
is  radiated.  The  amount  radiated  can  be  accounted 
for  and  the  formula  made  exactly  true. 

P rob.  75. — Suppose  we  have,  Fig.  23,  volume  </ =  7.2  cubic 
feet,  volume  a  =  .14  cubic  feet,  volume  c  =  1.08 
cubic  feet;  weight  steam  used  =  .35  pound ;  pres- 
sure c  =  loo,  pressure  a  =  15 ;  xe  =  -64,  xa  =  .9. 
What  should  theoretically  be  the  condition  of  the 
exhaust  steam  if  the  boiler  supplies  steam  having 
xf  =  i,  and  the  expansion  curve  is  a  rectangular 
hyperbola,  assuming  no  radiation  from  the  cylinder. 
The  heat  received  from  the  boiler  less  that  rejected  to 
the  condenser  or  air  is  the  work  done,  as  we  have 
assumed  no  radiation. 

The  heat  received  is  M(qc  +  rc). 
The  work  done  is 

100  x  144  x  .88  +  loo  x  144  x  i. 08  loge  —  -Q  —  7.06  x  144  x  15 

I  •  Oo 

=  27000  ft.-lbs.  =  34.7  h.  u. 
The  heat  rejected  is  M(qa-\-xara)  and 

M(qa  +  xara)  =  M(qc  +  rc}  -  34.7  ; 

•re  —  ga)  —  34-7 


Mra 

_  '35(297-9  +  884  —  181.8)  —  34.7  _ 
•35x965-1 

showing  tbat  under  these  conditions  the  exhaust 
steam  will  have  6.9  per  cent  moisture  in  it. 


NOTES   ON   THERMODYNAMICS.  69 

Prob,  76. — A  condensing  engine  working  between  150  and  4 
pounds  pressure  requires  15  pounds  of  dry  satu- 
rated steam  per  indicated  horse-power  per  hour. 
If  no  heat  is  radiated  from  the  cylinder,  what 
must  be  the  average  condition  of  the  exhaust  ? 

Prob.  77. — Draw  a  diagram  showing  the  quantity  of  dry  satu- 
rated steam  that  must  be  used  per  horse-power  per 
hour  in  order  that  the  exhaust  at  4  pounds  pres- 
sure may  be  dry  saturated  steam,  if  the  steam- 
pressure  is  80,  100,  120,  140,  and  160  pounds  per 
square  inch,  there  being  no  radiation. 


SHORT-TITLE    CATALOGUE 

OF  THE 

PUBLICATIONS 

OF 

JOHN   WILEY   &    SONS, 

NEW    YORK, 
LONDON:    CHAPMAN   &   HALL,  LIMITED. 


ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application. 

Books  marked  with  an  asterisk  are  sold  at  net  prices  only. 

All  books  are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

Armsby's  Manual  of  Cattle  Feeding 12mo,  $1  75 

Downing's  Fruit  and  Fruit  Trees 8vo,  5  00 

Grotenfelt's  The  Principles  of  Modern  Dairy  Practice.     (Woll.) 

12mo,  2  00 

Kemp's  Landscape  Gardening. 12mo,  2  50 

May uard' s  Landscape  Gardening 12rno,  1  50 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Woll's  Handbook  for  Farmers  and  Dairymen 12mo,  1  50 

ARCHITECTURE. 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Birkmire's  American  Theatres — Planning  and  Construction.  8  vo,  3  00 

„  "        Architectural  Iron  and  Steel .8vo,  3  50 

"        Compound  Riveted  Girders 8vo,  2  00 

"        Skeleton  Construction  in  Buildings 8vo,  3  00 

Planning  and  Construction  of  High  Office  Buildings. 

8vo,  3  50 

Briggs'  Modern  Am.  School  Building 8vo,  4  00 

Carpenter's  Heating  and  Ventilating  of  Buildings. 8vo,  3  00 

1 


Freitag's  Architectural  Engineering 8vo,  $2  50 

The  Fireproofing  of  Steel  Buildings : Svo,  250 

Gerhard's  Sanitary  House  Inspection 16mo;  1  00 

Theatre  Fires  and  Panics 12mo,  1  50 

Hatfield's  American  House  Carpenter 8vo,  5  00 

Holly's  Carpenter  and  Joiner 18mo,  75 

Kidder's  Architect  and  Builder's  Pocket-book. .  .16mo,  morocco,  4  00 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Monckton's  Stair  Building — Wood,  Iron,  and  Stone 4to,  4  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

"      Law  of  Operations  Preliminary  to  Construction  in  En- 
gineering and  Architecture. 8vo,  5  00 

Sheep,  5  50 

Worcester's  Small  Hospitals — Establishment  and  Maintenance, 
including  Atkinson's  Suggestions  for  Hospital  Archi- 
tecture  12mo,  125 

*  World's  Columbian  Exposition  of  1893 Large  4to,  1  00 

ARMY,  NAVY,  Etc. 

Bernadou's  Smokeless  Powder 12mo,  2  50 

*Bruffs  Ordnance  and  Gunnery 8vo,  6  00 

Chase's  Screw  Propellers 8vo,  3  00 

Cronkhite's  Gunnery  for  Non-com.  Officers 32mo,  morocco,  2  00 

*  Davis's  Treatise  on  Military  Law 8vo,  7  00 

Sheep,  7  50 

*  "       Elements  of  Law Svo,  250 

De  Brack's  Cavalry  Outpost  Duties.     (Carr.). . .  ,32mo,  morocco,  2  00 

Dietz's  Soldier's  First  Aid 16mo,  morocco,  1  25 

*  Dredge's  Modern  French  Artillery. . .  .Large  4to,  half  morocco,  15  00 

*  "          Record   of   the   Transportation    Exhibits    Building, 

World's  Columbian  Exposition  of  1893.. 4to,  half  morocco,  5  00 

Durand's  Resistance  and  Propulsion  of  Ships Svo,  5  00 

*  Fiebeger's    Field   Fortification,    including   Military   Bridges, 

Demolitions,  Encampments  and  Communications. 

Large  12mo,  2  00 

*Dyer's  Light  Artillery 12mo,  3  00 

*Hoff  s  Naval  Tactics Svo,  1  50 

*  Ingalls's  Ballistic  Tables Svo,  1  50 


Ingalls's  Handbook  of  Problems  in  Direct  Fire 8vo,  $4  00 

Mahan's  Permanent  Fortifications.  (Mercur.).8vo,  half  morocco,  7  50 

Manual  for  Courts-Martial 16mo,  morocco,  1  50 

*Mercur's  Attack  of  Fortified  Places 12mo,  2  00 

*  "        Elements  of  the  Art  of  War 8vo,  4  00 

Metcalfe's  Ordnance  and  Gunnery 12mo,  with  Atlas,  5  00 

Murray's  Infantry  Drill  Regulations  adapted  to  the  Springfield 

Rifle,  Caliber  .45 32mo,  paper,  10 

*  Phelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner 12mo,  4  00 

Sharpe's  Subsisting  Armies 32mo,  morocco,  1  50 

Wheeler's  Siege  Operations 8vo,  2  00 

Wiuthrop's  Abridgment  of  Military  Law 12mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene 16mo,  1  50 

Young's  Simple  Elements  of  Navigation  16mo,  morocco,  2  00 

first  edition 1  00 

ASSAYING. 

Fletcher's  Quant.  Assaying  with  the  Blowpipe..  16mo,  morocco,  1  50 

Furman's  Practical  Assaying 8vo,  3  00 

Kuuhardt's  Ore  Dressing .• 8vo,  1  50 

Miller's  Manual  of  Assaying 12mo,  1  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  00 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  00 

Thurston's  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Wilson's  Cyanide  Processes 12rno,  1  50 

The  Chlorination  Process 12uno,  150 

ASTRONOMY. 

Craig's  Azimuth, , 4to,  3  50 

Doolittle's  Practical  Astronomy 8vo,  4  00 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo.  3  00 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  00 

*  White's  Theoretical  and  Descriptive  Astronomy 12mo,  2  00 

BOTANY. 

Baldwin's  Orchids  of  New  England Small  8vo,  1  50 

Thome's  Structural  Botany 16mo,  2  25 

Westermaier's  General  Botany.     (Schneider.) 8vo,  2.  00 


BRIDGES,  ROOFS,   Etc. 

(See  also  ENGINEERING,  p.  7.) 

Boiler's  Highway  Bridges 8vo,  $2  00 

*     " '•    The  Thames  River  Bridge 4to,  paper,  500 

Burr's  Stresses  in  Bridges 8vo,  3  50 

Crehore's  Mechanics  of  the  Girder. 8vo,  5  00 

Du  Bois's  Stresses  in  Framed  Structures Small  4to,  10  00 

Foster's  Wooden  Trestle  Bridges  4to,  5  00 

Greene's  Arches  in  Wood,  etc 8vo,  2  50 

"         Bridge  Trusses 8vo,  2  50 

Roof  Trusses 8vo,  125 

Howe's  Treatise  on  Arches Svo,  4  00 

Johnson's  Modern  Framed  Structures Small  4to,  10  00 

Merrimau    &    Jacoby's     Text-book    of    Roofs     and    Bridges. 

Part  I.,  Stresses 8vo,  250 

Merriman    &    Jacoby's     Text-book    of    Roofs    and     Bridges. 

Part  II.,  Graphic  Statics 8vo,  2  50 

Merrimau    £    Jacoby's     Text-book    of    Roofs    and     Bridges. 

Part  III.,  Bridge  Design Svo,  2  50 

Merriman    &   Jacoby's    Text- book    of    Roofs    and    Bridges. 
Part  IV.,  Continuous,  Draw,  Cantilever,  Suspension,  and 

Arched  Bridges 8vo,  2  50 

*Morison's  The  Memphis  Bridge Oblong  4to,  10  00 

Waddell's  De  Pontibus  (a  Pocket-book  for  Bridge  Engineers). 

16mo.  morocco,  3  00 

"        Specifications  for  Steel  Bridges 12mo,  125 

Wood's  Construction  of  Bridges  and  Roofs Svo,  2  00 

Wright's  Designing  of  Draw  Spans.     Parts  I.  and  II..8vo,  each  2  50 

"      "           "         Complete 8vo,  350 

CHEMISTRY— BIOLOGY-PHARMACY— SANITARY  SCIENCE. 

Adriauce's  Laboratory  Calculations 12mo,  1  25 

Allen's  Tables  for  Iron  Analysis Svo,  3  00 

Austen's  Notes  for  Chemical  Students 12mo,  1  50 

Bolton's  Student's  Guide  in  Quantitative  Analysis Svo,  1  50 

Classen's  Analysis  by  Electrolysis.   (IIerrickandBo]twood.).8vo,  3  00 

4 


Cohn's  Indicators  and  Test-papers 12mo  $2  00 

Crafts's  Qualitative  Analysis.     (Schaeffer.) 12mo,  1  50 

Davenport's  Statistical  Methods  with  Special  Reference  to  Bio- 
logical Variations 12mo,  morocco,  1  25 

Drechsel's  Chemical  Reactions.    (Merrill.) 12mo,  1  25 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) 

12mo,  1  25 

Fresenius's  Quantitative  Chemical  Analysis.    (Allen.) 8vo,  6  00 

Qualitative         "              "           (Johnson.) 8vo,  300 

(Wells.)         Trans. 

16th  German  Edition 8vo,  5  00 

Fuertes's  Water  and  Public  Health 12mo,  1  50 

"         Water  Filtration  Works 12mo,  2  50 

Gill's  Gas  and  Fuel  Analysis 12mo,  125 

Goodrich's  Economic  Disposal  of  Towns'  Refuse Demy  8vo,  3  50 

Kammarsten's  Physiological  Chemistry.    (Maudel.) 8vo,  4  00 

Helm's  Principles  of  Mathematical  Chemistry.    (Morgan).  12mo,  1  50 

Hopkins'  Oil-Chemist's  Hand-book 8vo,  3  00 

Ladd's  Quantitative  Chemical  Analysis 12mo,  1  00 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  00 

Lob's  Electrolysis  and  Electrosyn thesis  of  Organic  Compounds, 

(Loreuz.) 12mo,  1  00 

Mandel's  Bio-chemical  Laboratory 12mo,  1  50 

Mason's  Water-supply 8vo,  5  00 

Examination  of  Water 12mo,  1  25 

Meyer's  Radicles  in  Carbon  Compounds.  (Tingle.) 12mo,  1  00 

Mixter's  Elementary  Text-book  of  Chemistry 12rno,  1  50 

Morgan's  The  Theory  of  Solutions  and  its  Results . .  .12mo,  1  00 

Elements  of  Physical  Chemistry 12mo,  2  00 

Nichols's  Water-supply  (Chemical  and  Sanitary) 8vo,  2  50 

O'Brine's  Laboratory  Guide  to  Chemical  Analysis 8vo,  2  00 

Pinner's  Organic  Chemistry.     (Austen.) 12mo,  1  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  00 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science..  12mo.  1  00 

and  Woodman's  Air,  Water,  and  Food 8vo,  2  00 

Ricketts   and   Russell's  Notes  on   Inorganic  Chemistry  (Non- 
metallic)  Oblong  8vo,  morocco,  75 

Rrtiears  Sewage  and  the  Bacterial  Purification  of  Sewage... 8vo,  3  50 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  00 

Schimpf  s  Volumetric  Analysis 12mo,  2  50 

Spencer's  Sugar  Manufacturer's  Handbook 16mo,  morocco,  2  00 

"          Handbook    for    Chemists    of   Beet    Sugar    Houses. 

16mo,  morocco,  3  00 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  Tillman's  Descriptive  General  Chemistry 8vo,  3  00 

5 


Van  Deventer's  Physical  Chemistry  for  Beginners.    (Boltwood.) 

12mo,  $1  50 

Wells'^  Inorganic  Qualitative  Analysis 12mo,  1  50 

"      Laboratory   Guide   in   Qualitative   Chemical   Analysis. 

8vo,  1  50 

Whipple's  Microscopy  of  Drinkiug-water 8vo,  3  50 

Wiechmann's  Chemical  Lecture  Notes 12rao,  3  00 

"            Sugar  Analysis , Small  8vo,  2  50 

Wulling's  Inorganic  Phar.  and  Med.  Chemistry 12mo,  2  00 

DRAWING. 

*  Bartlett's  Mechanical  Drawing 8vo,  3  00 

Hill's  Shades  and  Shadows  and  Perspective 8vo,  2  00 

MacCord's  Descriptive  Geometry 8vo,  3  00 

Kinematics 8vo,  500 

"          Mechanical  Drawing 8vo,  400 

Mahan's  Industrial  Drawing.    (Thompson.) 2  vols.,  8vo,  3  50 

Reed's  Topographical  Drawing.     (H.  A.) 4to,  5  00 

Reid's  A  Course  in  Mechanical  Drawing 8vo.  2  00 

"      Mechanical  Drawing  and  Elementary  Machine   Design. 

8vo,  3  00 

Smith's  Topographical  Drawing.     (Macmillan.) 8vo,  2  50 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

"        Drafting  Instruments 12mo,  1  25 

"        Free-hand  Drawing 12mo,  1  00 

"        Linear  Perspective 12mo,  100 

"        Machine  Construction 2  vols.,  8vo,  7  50 

Plane  Problems 12mo,  125 

"         Primary  Geometry 12mo,  75 

"        Problems  and  Theorems. 8vo,  250 

"        Projection  Drawing , 12mo,  1  50 

Shades  and  Shadows 8vo,  300 

"        Stereotomy — Stone-cutting 8vo,  250 

Whelpley's  Letter  Engraving 12mo,  2  00 

Wilson's  Free-hand  Perspective 8vo,  2  50 

ELECTRICITY,  MAGNETISM,  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.)   Small 

8vo,  3  00 

Anthony's  Theory  of  Electrical  Measurements 12mo,  1  00 

Barker's  Deep-sea  Soundings 8vo,  2  00 

Benjamin's  Voltaic  Cell 8vo,  3  00 

History  of  Electricity 8vo,  300 


Classen's  Analysis  by  Electrolysis.   (Heriick  and  Boltwood.)  8vo,  $3  00 
Crehore  and  Squier's  Experiments  with  a  New  Polarizing  Photo- 

Chronog-raph 8vo,  3  00 

Dawson's  Electric  Railways  and  Tramways.     Small,  4to,  half 

morocco,  12  50 

*  "  Engineering  "  and  Electric  Traction  Pocket-book.      16mo, 

morocco,  5  00 

*  Dredge's  Electric  Illuminations. .  .  .2  vols.,  4to,  half  morocco,  25  00 

Vol.11 4to,  750 

Gilbert's  De  magnete.     (Mottelay.) 8vo,  2  50 

Holman's  Precision  of  Measurements 8vo,  2  00 

"         Telescope-mirror-scale  Method Large  8vo,  75 

Le  Chatelier's  High  Temperatures.     (Burgess) 12mo,  3  00 

Lob's  Electrolysis  and  Electrosyn thesis  of  Organic  Compounds. 

(Lorenz.) 12mo,  100 

Lyous's  Electromagnetic  Phenomena  and  the  Deviations  of  the 

Compass 8vo,  6  00 

*Michie's  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  00 

Morgan's  The  Theory  of  Solutions  and  its  Results 12mo,  1  00 

Niaudet's  Electric  Batteries      (Fishback.) .12mo,  250 

*Parsball  &  Hobart  Electric  Generators.     Small  4to,  half  mor.,  10  00 

Pratt  and  Alden's  Street-railway  Road-beds 8vo,  2  00 

Reagan's  Steam  and  Electric  Locomotives 12mo,  2  00 

Thurston's  Stationary  Steam  Engines  for  Electric  Lighting  Pur- 
poses     8vo,  2  50 

*Tillman's  Heat 8vo,  1  50 

Tory  &  Pitcher's  Laboratory  Physics Small  8vo,  2  00 


ENGINEERING. 

CIVIL  —  MECHANICAL—  SANITARY,  ETC. 
(See  also    BRIDGES,    p.    4  ;   HYDRAULICS,   p.    9  ;   MATERIALS  OP  EN- 

GINEERING,   p.    11  ;    MECHANICS   AND   MACHINERY,  p.  12  ;    STEAM 

ENGINES  AND  BOILERS,  p.  14.) 


s  Masonry  Construction  .................  ..,«,,,...  8vo,  5  00 

"        Surveying  Instruments  .........  ................  12mo,  300 

Black's  U.  S.  Public  Works  ......................  Oblong  4to,  5  00 

Brooks's  Street-railway  Location  ......  ,  ........  16mo,  morocco,  1  50 

Butts's  Civil  Engineers'  Field  Book  ............  16mo,  morocco,  2  50 

Byrne's  Highway  Construction  .......................  ...  .8vo,  5  00 

"       Inspection  of  Materials  and  Workmanship  .......  16rno,  3  00 

Carpenter's  Experimental  Engineering  ...................  8vo,  6  00 

Church's  Mechanics  of  Engineering  —  Solids  and  Fluids.  ...8vo,  6  00 

7 


Church's  Notes  and  Examples  iii  Mechanics 8vo,     $2 

Crandall's  Earthwork  Tables 8vo,       1 

The  Transition  Curve IGmo,  morocco,       1 

Davis's  Elevation  and  Stadia  Tables Small  8vo,       1 

Dredge's     Penn.    Railroad     Construction,    etc.        Large     4to, 

half  morocco,  $10;  paper,       5 

*  Drinker's  Tunnelling 4to,  half  morocco,     25 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,       4 

Frizell's  Water  Power 8vo,       5 

Folwell's  Sewerage 8vo,       3 

"        Water-supply  Engineering .8vo,      4 

Fowler's  Coffer-dam  Process  for  Piers 8vo. 

Fuertes's  Water  Filtration  Works .12mo, 

Gerhard's  Sanitary  House  Inspection 12mo, 

Godwin's  Railroad  Engineer's  Field-book 16mo,  morocco, 

Goodrich's  Economic  Disposal  of  Towns'  Refuse Demy  8vo, 

Gore's  Elements  of  Geodesy 8vo, 

Hazlehurst's  Towers  and  Tanks  for  Cities  and  Towns 8vo, 

Howard's  Transition  Curve  Field-book 16mo,  morocco, 

Howe's  Retaining  Walls  (New  Edition.) 12mo, 

Hudson's  Excavation  Tables.     Vol.  II 8vo, 

Button's  Mechanical  Engineering  of  Power  Plants 8vo, 

"         Heat  and  Heat  Engines 8vo, 

Johnson's  Materials  of  Construction 8vo, 

"         Theory  and  Practice  of  Surveying Small  8vo, 

Kent's  Mechanical  Engineer's  Pocket-book 16mo,  morocco, 

Kiersted's  Sewage  Disposal 12mo,       1 

Mahan's  Civil  Engineering.      (Wood.) 8vo,       5 

Merriman  and  Brook's  Handbook  for  Surveyors. . .  .16mo,  mor.,       2 

Merriman's  Precise  Surveying  and  Geodesy 8vo,       2 

"          Sanitary  Engineering 8vo,       2 

Nagle's  Manual  for  Railroad  Engineers, ...... .16mo,  morocco,       3 

Ogdeu's  Sewer  Design 12mo,       2 

Patton's  Civil  Engineering 8vo,  half  morocco,       7 

"        Foundations , 8vo,       5 

Philbrick's  Field  Manual  for  Engineers 16mo,  morocco,       3 

Pratt  and  Alden's  Street-railway  Road-beds 8vo,      2 

Rockwell's  Roads  and  Pavements  in  France 12mo,       1 

Schuyler's  Reservoirs  for  Irrigation Large  8vo,       5 

Searles's  Field  Engineering , 16mo,  morocco,      3 

"       Railroad  Spiral 16mo,  morocco,       1 

Siebert  and  Biggin's  Modern  Stone  Cutting  and  Masonry. .  .8vo,       1 

Smart's  Engineering  Laboratory  Practice 12mo, 

Smith's  Wire  Manufacture  and  Uses Small  4to, 

Spalding's  Roads  and  Pavements 12mo, 

8 


Spalding's  Hydraulic  Cement 12mo,  $2  00 

Taylor's  Prismoiclal  Formulas  ami  Earthwork 8vo,  1  50 

Thurston's  Materials  of  Construction  „ 8vo,  5  00 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  00 

*  Trautwiue's  Civil  Engineer's  Pocket-book 16mo,  morocco,  5  00 

*  "           Cross-section Sheet,  25 

*  "           Excavations  and  Embankments 8vo,  2  00 

*  "            Laying  Out  Curves 12mo,  morocco,  2  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Waddell's  De  Pontibus  (A  Pocket-book  for  Bridge  Engineers). 

16mo,  morocco,  3  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

"      Law  of  Field  Operation  in  Engineering,  etc 8vo,  5  00 

Sheep,  5  50 

Warren's  Stereotomy— Stone-cutting 8vo,  2  50 

Webb's  Engineering  Instruments.  New  Edition.  16mo,  morocco,  1  25 

"      Railroad  Construction 8vo,  4  00 

Wegmanu's  Construction  of  Masonry  Dams 4to,  5  00 

Wellington's  Location  of  Railways Small  8vo,  5  00 

Wheeler's  Civil  Engineering 8vo,  4  00 

Wilson's  Topographical  Surveying 8vo,  3  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

HYDRAULICS. 

(See  also  ENGINEERING,  p.  7.) 
Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein. 

(Trautwiue.) 8vo,  2  00 

Bovey's  Treatise  on  Hydraulics. . . , 8vo,  4  00 

Church's  Mechanics  of  Engineering,  Solids,  and  Fluids 8vo,  6  00 

Coffin's  Graphical  Solution  of  Hydraulic  Problems 12mo,  2  50 

Ferrers  Treatise  on  the  Winds,  Cyclones,  and  Tornadoes. .  .8vo,  4  00 

Fol well's  Water  Supply  Engineering 8vo,  4  00 

Frizell's  Water-power 8vo,  5  00 

Fuertes's  Water  and  Public  Health 12mo,  1  50 

Water  Filtration  Works 12mo.  250 

Ganguillet  &  Kutter's  Flow  of  Water.     (Heriug  &  Trautwine.) 

8vo,  4  00 

Hazen's  Filtration  of  Public  Water  Supply 8vo,  3  00 

Hazlehurst's  Towers  and  Tanks  for  Cities  and  Towns 8vo,  2  50 

Herschel's  1 15  Experiments  8vo,  2  00 

Kiersted's  Sewage  Disposal 12mo,  1  25 

Mason's  Water  Supply 8vo,  5  00 

"    Examination  of  Water 12mo,  1  25 

Merriman's  Treatise  on  Hydraulics 8vo,  4  00 

9 


Nichols's  Water  Supply  (Chemical  and  Sanitary) 8vo,  $2  50 

Schuyler's  Reservoirs  for  Irrigation Large  8vo,  5  00 

Turueaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Wegmann's  Water  Supply  of  the  City  of  New  York 4to,  10  00 

Weisbach's  Hydraulics.     (Du  Bois.) 8vo,  5  00 

Whipple's  Microscopy  of  Drinking  Water 8vo,  3  50 

Wilson's  Irrigation  Engineering. . : 8vo,  4  00 

Hydraulic  and  Placer  Mining 12mo,  2  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Theory  of  Turbines 8vo,  2  50 

LAW. 

Davis's  Elements  of  Law 8vo,  2  50 

Treatise  on  Military  Law 8vo,  7  00 

Sheep,  7  50 

Manual  for  Courts-martial 16mo,  morocco,  1  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

"      Law  of  Contracts 8vo,  3  00 

"      Law  of  Operations  Preliminary  to  Construction  in  En- 
gineering and  Architecture 8vo,  5  00 

Sheep,  5  50 

Winthrop's  Abridgment  of  Military  Law 12mo,  2  50 

MANUFACTURES. 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Beaumont's  Woollen  and  Worsted  Manufacture 12mo,  1  50 

Bollaud's  Encyclopaedia  of  Founding  Terms 12mo,  3  00 

"         The  Iron  Founder 12mo,  250 

Supplement 12mo,  250 

Eissler's  Explosives,  Nitroglycerine  and  Dynamite 8vo,  4  00 

Ford's  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

Metcalf s  Steel— A  Manual  for  Steel  Users 12mo,  2  00 

*  Reisig's  Guide  to  Piece  Dyeing Svo,  25  00 

Spencer's  Sugar  Manufacturer's  Handbook 16mo,  morocco,  2  00 

"        Handbook    for    Chemists    of    Beet    Sugar    Houses. 

16mo,  morocco,  3  00 

Thurston's  Manual  of  Steam  Boilers Svo,  5  00 

Walke's  Lectures  on  Explosives Svo,  4  00 

West's  American  Foundry  Practice 12mo,  2  50 

"      Moulder's  Text-book  12mo,  2  50 

Wiechmaun's  Sugar  Analysis Small  8vo,  2  50 

Woodbury's  Fire  Protection  of  Mills Svo,  9,  50 

10 


MATERIALS  OF  ENGINEERING. 

(See  also  ENGINEERING,  p.  7.) 

Baker's  Masonry  Construction 8vo,  $5  00 

Bovey's  Strength  of  Materials 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  Materials Svo,  5  00 

Byrne's  Highway  Construction Svo,  5  00 

Church's  Mechanics  of  Engineering — Solids  and  Fluids Svo,  6  00 

Du  Bois's  Stresses  in  Framed  Structures Small  4to,  10  00 

Johnson's  Materials  of  Construction Svo,  6  00 

Lanza's  Applied  Mechanics Svo,  7  50 

Marlens's  Testing  Materials.     (Henning.) 2  vols.,  Svo,  7  50 

Merrill's  Stones  for  Building  and  Decoration Svo,  5  00 

Merriman's  Mechanics  of  Materials Svo,  4  00 

' '           Strength  of  Materials 12mo,  1  00 

Pattou's  Treatise  on  Foundations Svo,  5  00 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Spalding's  Roads  and  Pavements 12mo,  2  00 

Thurston's  Materials  of  Construction Svo,  5  00 

Materials  of  Engineering 3  vols.,  Svo,  8  00 

Vol.  I,  Non-metallic  .., 8 .-,  200 

Vol.  II. ,  Iron  and  Steel Svo,  3  50 

Vol.  III.,  Alloys,  Brasses,  and  Bronzes Svo,  2  50 

Wood's  Resistance  of  Materials Svo,  2  00 

MATHEMATICS. 

Baker's  Elliptic  Functions Svo,  1  50 

*Bass's  Differential  Calculus 12mo,  4  00 

Briggs's  Plane  Analytical  Geometry 12mo,  1  00 

Chapman's  Theory  of  Equations 12mo,  1  50 

Compton's  Logarithmic  Computations 12nio,  1  50 

Davis's  Introduction  to  the  Logic  of  Algebra Svo,  1  50 

Halsted's  Elements  of  Geometry t..8vo,  1  75 

"        Synthetic  Geometry Svo,  150 

Johnson's  Curve  Tracing 12mo,  1  00 

"        Differential  Equations— Ordinary  and  Partial. 

Small  Svo,  3  50 

*"        Integral  Calculus 12mo,  150 

11  "  "        Unabridged.     Small  Svo.    (In  press.) 

"        Least  Squares 12mo,  1  50 

*Ludlow's  Logarithmic  and  Other  Tables.     (Bass.) Svo,  2  00 

*      "        Trigonometry  with  Tables.     (Bass.) Svo,  3  00 

*Mahan's  Descriptive  Geometry  (Stone  Cutting)  .Svo,  1  50 

Merrimau  and  Woodward's  Higher  Mathematics Svo,  5  00 

11 


Merrirnan's  Method  of  Least  Squares £vo,  $%  00 

Rice  and  Johnson's  Differential  and  Integral  Calculus, 

2  vols.  in  1,  small  8vo,  2  50 

"                  Differential  Calculus Small  8vo,  3  00 

"  Abridgment  of  Differential  Calculus. 

Small  8vo,  1  50 

Totten's  Metrology 8vo,  2  50 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

41         Drafting  Instruments 12mo,  1  25 

"        Free-hand  Drawing 12mo,  1  00 

"        Linear  Perspective 12mo,  1  00 

"        Primary  Geometry 12mo,  75 

Plane  Problems 12rno,  1  25 

"        Problems  and  Theorems 8vo,  2  50 

"        Projection  Drawing 12mo,  1  50 

Wood's  Co-ordinate  Geometry 8vo,  2  00 

Trigonometry 12mo,  100 

Woolf's  Descriptive  Geometry Large  8vo,  3  00 

MECHANICS-MACHINERY. 

(See  also  ENGINEERING,  p.  7.) 

Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

Benjamin's  Wrinkles  and  Recipes 12mo,  2  00 

Chordal's  Letters  to  Mechanics 1 2mo,  2  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

"        Notes  and  Examples  in  Mechanics 8vo,  2  00 

Crehore's  Mechanics  of  the  Girder 8vo,  5  00 

Cromwell's  Belts  and  Pulleys 12mo,  1  50 

Toothed  Gearing 12mo,  150 

Compton's  First  Lessons  in  Metal  Working 12mo,  1  50 

Compton  and  De  Groodt's  Speed  Lathe 12mo,  1  50 

Dana's  Elementary  Mechanics 12mo,  1  50 

Dingey's  Machinery  Pattern  Making 12nio,  2  00 

*  Dredge's     Trans.     Exhibits    Building,     World     Exposition. 

Large  4to,  half  morocco,  5  00 

Du  Bois's  Mechanics.     Vol.  I.,  Kinematics , 8vo,  3  50 

Vol.  II.,  Statics 8vo,  400 

Vol.  III.,  Kinetics 8vo,  350 

Fitzgerald's  Boston  Machinist 18mo,  100 

Flather's  Dynamometers 12mo,  200 

"        Rope  Driving 12mo,  200 

Hall's  Car  Lubrication 12mo,  1  00 

Holly's  Saw  Filing 18mo,  75 

12 


*  Johnson's  Theoretical  Mechanics.     An  Elementary  Treatise. 

12mo,  $3  00 

Joues's  Machine  Design.     Part  I.,  Kinematics 8vo,  1  50 

"  Part  II.,  Strength  and  Proportion  of 

Machine  Parts 8vo,  3  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

MacCord's  Kinematics 8vo,  5  00 

Merriman's  Mechanics  of  Materials 8vo,  4  00 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

*Michie's  Analytical  Mechanics 8vo,  4  00 

Richards's  Compressed  Air 12mo,  1  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Thurstou's  Friction  and  Lost  Work 8vo,  3  00 

The  Animal  as  a  Machine 12mo,  1  00 

Warren's  Machine  Construction 2  vols.,  8vo,  7  50 

Weisbach's  Hydraulics  and  Hydraulic  Motors.    (Du  Bois.)..8vo,  5  00 
"          Mechanics    of    Engineering.      Vol.    III.,    Part   I., 

Sec.  I.     (Klein.) 8vo,  500 

Weisbach's  Mechanics    of  Engineering.     Vol.   III.,    Part  I., 

Sec.  II.     (Klein.) 8vo,  500 

Weisbach's  Steam  Engines.     (Du  Bois.) 8vo,  5  00 

Wood's  Analytical  Mechanics 8vo,  3  00 

"      Elementary  Mechanics 12mo,  125 

"               "                 "           Supplement  and  Key 12rno,  125 

METALLURGY. 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Egleston's  Gold  and  Mercury Large  8vo,  7  50 

"         Metallurgy  of  Silver Large  8vo,  7  50 

*  Kerl's  Metallurgy— Steel,  Fuel,  etc 8vo,  15  00 

Kunhardt's  Ore  Dressing  in  Europe 8vo,  1  50 

Metcalf's  Steel—A  Manual  for  Steel  Users 12mo,  2  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  00 

Thurstou's  Iron  and  Steel 8vo,  3  50 

Alloys 8vo,  250 

Wilson's  Cyanide  Processes 12mo,  1  50 

MINERALOGY   AND  MINING. 

Barringer's  Minerals  of  Commercial  Value. .  ..Oblong  morocco,  2  50 

Beard's  Ventilation  of  Mines 12mo,  2  50 

Boyd's  Resources  of  South  Western  Virginia 8vo,  3  00 

"      Map  of  South  Western  Virginia Pocket-book  form,  2  00 

Brush  and  Penfield's  Determinative  Mineralogy.    New  Ed.  8vo,  4  00 

13 


Chester's  Catalogue  of  Minerals 8vo,  $1  25 

Paper,  50 

Dictionary  of  the  Names  of  Minerals 8vo,  3  00 

Dana's  American  Localities  of  Minerals Large  8vo,  1  00 

"      Descriptive  Mineralogy.  (E.S.)  Large  8vo.  half  morocco,  12  50 

"      First  Appendix  to  System  of  Mineralogy Large  8vo,  1  00 

"      Mineralogy  and  Petrography.     (J.  D. ) 12mo,  2  00 

"      Minerals  and  How  to  Study  Them.     (E.  S.).. 12mo,  1  50 

4<      Text-book  of  Mineralogy.     (E.  S.)..  .New  Edition.     8vo,  400 

*  Drinker's  Tunnelling,  Explosives,  Compounds,  and  Rock  Drills. 

4to,  half  morocco,  25  00 

Eglestou's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,  4  00 

Hussak's  Rock  forming  Minerals.     (Smith.) Small  Svo,  2  00 

Ihlseng's  Manual  of  Mining. .    . .   Svo,  4  00 

Kunhardt's  Ore  Dressing  in  Europe , 8vo,  1  50 

O'Driscoll's  Treatment  of  Gold  Ores Svo,  2  00 

*  Penfield's  Record  of  Mineral  Tests Paper,  8vo,  50 

Rosenbusch's    Microscopical    Physiography   of    Minerals    and 

Rocks.     (Iddings.) Svo,  500 

Sawyer's  Accidents  in  Mines Large  Svo,  7  00 

Stockbridge's  Rocks  and  Soils Svo,  2  50 

*Tillman's  Important  Minerals  and  Rocks Svo,  2  00 

"Walke's  Lectures  on  Explosives Svo,  4  00 

Williams's  Lithology Svo,  3  00 

Wilson's  Mine  Ventilation 12ino,  125 

"        Hydraulic  and  Placer  Mining 12mo,  250 

STEAM  AND  ELECTRICAL  ENGINES,  BOILERS,  Etc. 

(See  also  ENGINEERING,  p.  7.) 

Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Clerk's  Gas  Engine < Small  Svo,  4  00 

Ford's  Boiler  Making  for  Boiler  Makers ISmo,  1  00 

Hemenway's  Indicator  Practice 12mo,  2  00 

Kent's  Steam-boiler  Economy , 8vo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector Svo,  1  50 

MacCord's  Slide  Valve Svo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabodyand  Miller's  Steam-boilers .Svo,  4  00 

Peabody's  Tables  of  Saturated  Steam Svo,  1  00 

"          Thermodynamics  of  the  Steam  Engine Svo,  5  00 

"          Valve  Gears  for  the  Steam  Engine Svo,  2  50 

"           Manual  of  the  Steam-engine  Indicator 12mo,  1  50 

Fray's  Twenty  Years  with  the  Indicator. ...,....,.  .Large  Svo,  2  50 

14 


Pupin  and  Osterberg's  Thermodynamics 12mo,  $1  25 

Reagan's  Steam  and  Electric  Locomotives 12mo,  2  00 

Rontgen's  Thermodynamics.     (Du  Bois.) Svo,  5  00 

Sinclair's  Locomotive  Running 12mo,  2  00 

Snow's  Steam-boiler  Practice Svo.  3  00 

Thurston's  Boiler  Explosions 12mo,  1  50 

Engine  and  Boiler  Trials Svo,  500 

"  Manual  of  the  Steam  Engine.      Part  I.,  Structure 

and  Theory Svo,  6  00 

•'  Manual  of   the    Steam  Engine.      Part  II.,  Design, 

Construction,  and  Operation Svo,  6  00 

2  parts,  10  00 

Philosophy  of  the  Steam  Engine 12mo,  75 

"  Reflection  on  the  Motive  Power  of  Heat.    (Caruot.) 

12mo,  1  50 

"           Stationary  Steam  Engines  Svo,  2  50 

"           Steam-boiler  Construction  and  Operation Svo,  5  00 

Spaugler's  Valve  Gears Svo,  2  50 

Notes  on  Thermodynamics 12mo,  1  00 

Weisbach's  Steam  Engine.     (Du  Bois.) . Svo,  5  00 

Whitbam's  Steam-engine  Design..,, Svo,  5  00 

Wilson's  Steam  Boilers.     (Flather.)       12mo,  250 

Wood's  Thermodynamics,  Heat  Motors,  etc Svo,  4  00 

TABLES,  WEIGHTS,  AND  MEASURES. 

Adriance's  Laboratory  Calculations 12mo,  1  25 

Allen's  Tables  for  Iron  Analysis Svo,  3  00 

Bixby's  Graphical  Computing  Tables Sheet,  25 

Comptou's  Logarithms 12mo,  1  50 

Crandall's  Railway  and  Earthwork  Tables Svo,  1  50 

Davis's  Elevation  and  Stadia  Tables Small  Svo,  1  00 

Fisher's  Table  of  Cubic  Yards .Cardboard,  25 

Hudson's  Excavation  Tables.     Vol.  II Svo,  1  00 

Johnson's  Stadia  and  Earthwork  Tables , Svo,  1  25 

Ludlow's  Logarithmic  and  Other  Tables.     (Bass.) 12mo,  2  00 

Totteu's  Metrology Svo,  2  50 

VENTILATION. 

Baldwin's  Steam  Heating 12rno,  2  50 

Beard's  Ventilation  of  Mines 12mo,  2  50 

Carpenter's  Heating  and  Ventilating  of  Buildings Svo,  3  00 

Gerhard's  Sanitary  House  Inspection 12mo,  1  00 

Wilson's  Mine  Ventilation 12mo,  1  25 

15 


MISCELLANEOUS  PUBLICATIONS. 

Alcott's  Gems,  Sentiment,  Language Gilt  edges,  $5  00 

Emmon's  Geological  Guide-book  of  the  Rocky  Mountains.  .8vo,  1  50 

Ferret's  Treatise  on  the  Winds 8vo,  4  00 

Haines's  Addresses  Delivered  before  the  Am.  Ry.  Assn.  ..12mo,  2  50 

Mott's  The  Fallacy  of  the  Present  Theory  of  Sound.  ,Sq.  IGnio,  1  00 

Richards's  Cost  of  Living 12mo,  1  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute .8vo,  3  00 

Rotherham's    The    New    Testament    Critically    Emphasized. 

12rno,  1  50 
"              The  Emphasized  New  Test.     A  new  translation. 

Large  8vo,  2  00 

Totten's  An  Important  Question  in  Metrology 8vo,  8  50 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

FOR  SCHOOLS  AND  THEOLOGICAL  SEMINARIES. 

Gesenius's  Hebrew  and   Chaldee  Lexicon  to  Old   Testament. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Green's  Elementary  Hebrew  Grammar 12mo,  1  25 

"       Grammar  of  the  Hebrew  Language  (New  Edition).  8  vo,  3  00 

"       Hebrew  Chrestomathy 8vo,  2  00 

Letteris's   Hebrew  Bible  (Massoretic  Notes  in  English). 

8vo,  arabesque,  2  25 

MEDICAL. 

Hammarsten's  Physiological  Chemistry.   (Mandel.) 8vo,      4  00 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food. 

Large  mounted  chart,       1  25 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,      2  00 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,       3  50 

Woodhull's  Military  Hygiene , 16mo,      1  50 

"Worcester's  Small  Hospitals — Establishment  and  Maintenance, 
including  Atkinson's  Suggestions  for  Hospital  Archi- 
tecture  12mo,  1  25 

16 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


280ct'583B 


OCT  2  8  1959 


LD21-100m-7,'33 


VB  09754 


1 

di 


96050 


